Data Camp Insights
davegoblue
April 26, 2017
Background and Overview
DataCamp offer interactive courses related to R Programming. While some is review, it is helpful to see other perspectives on material. As well, DataCamp has some interesting materials on packages that I want to learn better (ggplot2, dplyr, ggvis, etc.). This document summarizes a few key insights from:
This document is currently split between _v003 and v_003_a due to the need to keep the number of DLL that it opens below the hard-coded maximum. This introductory section needs to be re-written, and the contents consolidated, at a future date.
- R Programming (Introductory R, Intermediate R, Writing Functions in R, Object Oriented Programming in R, Introduction to Tidyverse)
- Importing and Cleaning Data (Cleaning Data in R, Importing Data in to R)
- Data Manipulation (dplyr, data.table, xts/zoo, dplyr joins, cases for EDA/Time Series/Pitch Data)
- Data Visualization (base, ggplot2 parts I/II/III, ggvis, geospatial)
- Statistics (8 refresher modules)
- Machine Learning (3 modules + 1 text mining case)
- R Studio (2 parts) and R Markdown (1 module)
The original DataCamp_Insights_v001 and DataCamp_Insights_v002 documents have been split for this document:
- This DataCamp_Insights_v003 document contains evolving sections on R Programming, Machine Learning, and RStudio / R Markdown
- Importing and Cleaning Data components have been moved to DataCamp_ImportClean_v002
- Data Manipulation components have been moved to DataCamp_DataManipulation_v002
- Visualization components have been moved to DataCamp_Visualization_v002
- Statistics components have been moved to DataCamp_Statistics_v002
Working with Dates and Times in R
Chapter 1 - Dates and Times in R
Introduction to dates - including the built-in methods for R:
- Differences in M-D-Y and D-M-Y
- ISO8601 is a standard for dates - components should be decreasing such as YYYY-MM-DD
- The numbers should all be padded with leading zeroes
- A separator is not required, but it must be a dash (-) if used
- The numbers should all be padded with leading zeroes
- R will generally require input using as.Date(
)
- Some functions that read in data will automatically recognize and parse dates in a variety of formats
- In particular the import functions, like read_csv(), in the readr package will recognize dates in a few common formats
- There is also the anytime() function in the anytime package whose sole goal is to automatically parse strings as dates regardless of the format
- In particular the import functions, like read_csv(), in the readr package will recognize dates in a few common formats
Why use dates?
- Behind the scenes, dates are stored as the number of days since 1970-01-01
- Can compare dates, take differences of dates, use dates for plotting, and the like
- Can compare dates, take differences of dates, use dates for plotting, and the like
- R releases have a major, minor, and patch
- Patch starts at zero with a new minor and increments by 1
- Minor starts at zero with a new major and incerements by 1
- Patch starts at zero with a new minor and increments by 1
What about times?
- R also has the built-in capability to handle datetimes
- ISO8601 has standards for datetimes also - YYYYMMDD HH:MM:SS
- Two capabilities for storing times in R
- POSIXlt - list with named components
- POSIXct - seconds since 1970-01-01 00:00:00 (typically better for data frames, and focus of this module)
- POSIXlt - list with named components
- Can convert to POSIXct using as.POSIXct(
)
- Can pass a timezone, and the default assumption is local time
- If the string is passed as YYYYMMDDTHH:MM:SSZ then the assumption is made of Zulu (UTC) time
- If the string is passed as YYYYMMDDTHH:MM:SSZ then the assumption is made of Zulu (UTC) time
- One drawback is that as.POSIXct() does not naturally recognize the timezones, so some additional work is required to properly enter a datetime
Why lubridate?
- The lubridate package is designed to make it easier to work with dates and times
- Part of the tidyverse - designed for humans, and integrates nicely to data analysis pipelines
- Consistent behavior regardless of the underlying objects
- Part of the tidyverse - designed for humans, and integrates nicely to data analysis pipelines
- Easier to use, and more forgiving of formats
- Has capability for time spans (time between two times, such as time for reign of monarchs)
Example code includes:
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(ggplot2)
# The date R 3.0.0 was released
x <- "2013-04-03"
# Examine structure of x
str(x)## chr "2013-04-03"# Use as.Date() to interpret x as a date
x_date <- as.Date(x)
# Examine structure of x_date
str(x_date)## Date[1:1], format: "2013-04-03"# Store April 10 2014 as a Date
april_10_2014 <- as.Date("2014-04-10")
# Load the readr package
library(readr)
# Use read_csv() to import rversions.csv
releases <- read_csv("./RInputFiles/rversions.csv")## Parsed with column specification:
## cols(
## major = col_integer(),
## minor = col_integer(),
## patch = col_integer(),
## date = col_date(format = ""),
## datetime = col_datetime(format = ""),
## time = col_time(format = ""),
## type = col_character()
## )# Examine the structure of the date column
str(releases$date)## Date[1:105], format: "1997-12-04" "1997-12-21" "1998-01-10" "1998-03-14" ...# Load the anytime package
library(anytime)
# Various ways of writing Sep 10 2009
sep_10_2009 <- c("September 10 2009", "2009-09-10", "10 Sep 2009", "09-10-2009")
# Use anytime() to parse sep_10_2009
anytime(sep_10_2009)## [1] "2009-09-10 CDT" "2009-09-10 CDT" "2009-09-10 CDT" "2009-09-10 CDT"# Set the x axis to the date column
ggplot(releases, aes(x = date, y = type)) +
geom_line(aes(group = 1, color = factor(major)))
# Limit the axis to between 2010-01-01 and 2014-01-01
ggplot(releases, aes(x = date, y = type)) +
geom_line(aes(group = 1, color = factor(major))) +
xlim(as.Date("2010-01-01"), as.Date("2014-01-01"))## Warning: Removed 87 rows containing missing values (geom_path).
# Specify breaks every ten years and labels with "%Y"
ggplot(releases, aes(x = date, y = type)) +
geom_line(aes(group = 1, color = factor(major))) +
scale_x_date(date_breaks = "10 years", date_labels = "%Y")
# Find the largest date
last_release_date <- max(releases$date)
# Filter row for last release
last_release <- filter(releases, date == last_release_date)
# Print last_release
last_release## # A tibble: 1 x 7
## major minor patch date datetime time type
## <int> <int> <int> <date> <dttm> <time> <chr>
## 1 3 4 1 2017-06-30 2017-06-30 07:04:11 07:04 patch# How long since last release?
Sys.Date() - last_release_date## Time difference of 280 days# Use as.POSIXct to enter the datetime
as.POSIXct("2010-10-01 12:12:00")## [1] "2010-10-01 12:12:00 CDT"# Use as.POSIXct again but set the timezone to `"America/Los_Angeles"`
as.POSIXct("2010-10-01 12:12:00", tz = "America/Los_Angeles")## [1] "2010-10-01 12:12:00 PDT"# Use read_csv to import rversions.csv
releases <- read_csv("./RInputFiles/rversions.csv")## Parsed with column specification:
## cols(
## major = col_integer(),
## minor = col_integer(),
## patch = col_integer(),
## date = col_date(format = ""),
## datetime = col_datetime(format = ""),
## time = col_time(format = ""),
## type = col_character()
## )# Examine structure of datetime column
str(releases$datetime)## POSIXct[1:105], format: "1997-12-04 08:47:58" "1997-12-21 13:09:22" ...# Import "cran-logs_2015-04-17.csv" with read_csv()
logs <- read_csv("./RInputFiles/cran-logs_2015-04-17.csv")## Parsed with column specification:
## cols(
## datetime = col_datetime(format = ""),
## r_version = col_character(),
## country = col_character()
## )# Print logs
logs## # A tibble: 100,000 x 3
## datetime r_version country
## <dttm> <chr> <chr>
## 1 2015-04-16 22:40:19 3.1.3 CO
## 2 2015-04-16 09:11:04 3.1.3 GB
## 3 2015-04-16 17:12:37 3.1.3 DE
## 4 2015-04-18 12:34:43 3.2.0 GB
## 5 2015-04-16 04:49:18 3.1.3 PE
## 6 2015-04-16 06:40:44 3.1.3 TW
## 7 2015-04-16 00:21:36 3.1.3 US
## 8 2015-04-16 10:27:23 3.1.3 US
## 9 2015-04-16 01:59:43 3.1.3 SG
## 10 2015-04-18 15:41:32 3.2.0 CA
## # ... with 99,990 more rows# Store the release time as a POSIXct object
release_time <- as.POSIXct("2015-04-16 07:13:33", tz = "UTC")
# When is the first download of 3.2.0?
logs %>%
filter(r_version == "3.2.0")## # A tibble: 35,928 x 3
## datetime r_version country
## <dttm> <chr> <chr>
## 1 2015-04-18 12:34:43 3.2.0 GB
## 2 2015-04-18 15:41:32 3.2.0 CA
## 3 2015-04-18 14:58:41 3.2.0 IE
## 4 2015-04-18 16:44:45 3.2.0 US
## 5 2015-04-18 04:34:35 3.2.0 US
## 6 2015-04-18 22:29:45 3.2.0 CH
## 7 2015-04-17 16:21:06 3.2.0 US
## 8 2015-04-18 20:34:57 3.2.0 AT
## 9 2015-04-17 18:23:19 3.2.0 US
## 10 2015-04-18 03:00:31 3.2.0 US
## # ... with 35,918 more rows# Examine histograms of downloads by version
ggplot(logs, aes(x = datetime)) +
geom_histogram() +
geom_vline(aes(xintercept = as.numeric(release_time)))+
facet_wrap(~ r_version, ncol = 1)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Chapter 2 - Parsing and Manipulating Dates with lubridate
Parsing dates with lubridate:
- lubridate::ymd() will manage dates in format ymd, even if they are not properly ISO formatted (have separators, English abbreviations, and the like)
- Analogous behaviors from ydm(), mdy(), myd(), dmy(), dym(), day_hm()
- Assumes UTC unless otherwise specified
- All the functions with y, m and d in any order exist
- If your dates have times as well, you can use the functions that start with ymd, dmy, mdy or ydm and are followed by any of _h, _hm or _hms
- To see all the functions available look at ymd() for dates and ymd_hms() for datetimes
- Analogous behaviors from ydm(), mdy(), myd(), dmy(), dym(), day_hm()
- lubridate::parse_date_time(x=, orders=)
- The orders = argument is a sequence of characters, reflecting the order in the input
- y-year with century, Y-year without century, m-month, d-day, H-hours (24-hour), M-minutes, S-seconds, and many others
- a-abbreviated weekday, A-full weekday, b-abbreviate month, B-full month, I-hours (12-hour), p-AM/PM, z-timezone (offset in minutes/seconds from UTC)
- Can pass a vector of sequences to orders=, such as orders=c(“ymd”, “dmy”), if some of the dates are formatted differently than others
- The orders = argument is a sequence of characters, reflecting the order in the input
Weather in Auckland (data from Weather Underground, METAR from Auckland airport):
- Data are available in akl_weather_daily.csv and akl_weather_hourly_2016.csv
- The lubridate::make_date(year, month, date) will produce a date from its components (these components can be vectors, such as columns in a frame
- There is also a lubridate::make_datetime(year, month, day, hour, min, sec)
Extracting parts of a datetime:
- The lubridate::year() will pull out the year from a datetime object
- month(), day(), hour(), minute(), second() will do the same
- wday() is the weekday (1-7), while yday() is the Julian date (1-366) and tz() is the timezone
- month(), day(), hour(), minute(), second() will do the same
- The extractors can also be used to set a component of the datetime object
- Several functions return booleans, more or less answers to “is this a” questions
- leap_year(), am(), pm(), dst(), quarter() will return 1-4, semester() will return 1-2
- Months of course are different lengths so we should really correct for that, take a look at days_in_month() for helping with that
- leap_year(), am(), pm(), dst(), quarter() will return 1-4, semester() will return 1-2
Rounding datetimes:
- The lubridate::floor_date(unit=) will round-down to the requested unit, such as “hour”
- round_date() for nearest
- ceiling_date() for round-up
- round_date() for nearest
- Units can be specified as “second”, “minute”, “hour”, “day”, “week”, “month”, “bimonth”, “quarter”, “halfyear”, “year”
Example code includes:
library(lubridate)##
## Attaching package: 'lubridate'## The following object is masked from 'package:base':
##
## datelibrary(readr)
library(dplyr)
library(ggplot2)
library(ggridges)
library(stringr)
# Parse x
x <- "2010 September 20th" # 2010-09-20
ymd(x)## [1] "2010-09-20"# Parse y
y <- "02.01.2010" # 2010-01-02
dmy(y)## [1] "2010-01-02"# Parse z
z <- "Sep, 12th 2010 14:00" # 2010-09-12T14:00
mdy_hm(z)## [1] "2010-09-12 14:00:00 UTC"# Specify an order string to parse x
x <- "Monday June 1st 2010 at 4pm"
parse_date_time(x, orders = "AmdyIp")## [1] "2010-06-01 16:00:00 UTC"# Specify order to include both "mdy" and "dmy"
two_orders <- c("October 7, 2001", "October 13, 2002", "April 13, 2003",
"17 April 2005", "23 April 2017")
parse_date_time(two_orders, orders = c("mdy", "dmy"))## [1] "2001-10-07 UTC" "2002-10-13 UTC" "2003-04-13 UTC" "2005-04-17 UTC"
## [5] "2017-04-23 UTC"# Specify order to include "dOmY", "OmY" and "Y"
short_dates <- c("11 December 1282", "May 1372", "1253")
parse_date_time(short_dates, orders = c("dOmY", "OmY", "Y"))## [1] "1282-12-11 UTC" "1372-05-01 UTC" "1253-01-01 UTC"# Import CSV with read_csv()
akl_daily_raw <- read_csv("./RInputFiles/akl_weather_daily.csv")## Parsed with column specification:
## cols(
## date = col_character(),
## max_temp = col_integer(),
## min_temp = col_integer(),
## mean_temp = col_integer(),
## mean_rh = col_integer(),
## events = col_character(),
## cloud_cover = col_integer()
## )# Print akl_daily_raw
akl_daily_raw## # A tibble: 3,661 x 7
## date max_temp min_temp mean_temp mean_rh events cloud_cover
## <chr> <int> <int> <int> <int> <chr> <int>
## 1 2007-9-1 60 51 56 75 <NA> 4
## 2 2007-9-2 60 53 56 82 Rain 4
## 3 2007-9-3 57 51 54 78 <NA> 6
## 4 2007-9-4 64 50 57 80 Rain 6
## 5 2007-9-5 53 48 50 90 Rain 7
## 6 2007-9-6 57 42 50 69 <NA> 1
## 7 2007-9-7 59 41 50 77 <NA> 4
## 8 2007-9-8 59 46 52 80 <NA> 5
## 9 2007-9-9 55 50 52 88 Rain 7
## 10 2007-9-10 59 50 54 82 Rain 4
## # ... with 3,651 more rows# Parse date
akl_daily <- akl_daily_raw %>%
mutate(date = ymd(date))
# Print akl_daily
akl_daily## # A tibble: 3,661 x 7
## date max_temp min_temp mean_temp mean_rh events cloud_cover
## <date> <int> <int> <int> <int> <chr> <int>
## 1 2007-09-01 60 51 56 75 <NA> 4
## 2 2007-09-02 60 53 56 82 Rain 4
## 3 2007-09-03 57 51 54 78 <NA> 6
## 4 2007-09-04 64 50 57 80 Rain 6
## 5 2007-09-05 53 48 50 90 Rain 7
## 6 2007-09-06 57 42 50 69 <NA> 1
## 7 2007-09-07 59 41 50 77 <NA> 4
## 8 2007-09-08 59 46 52 80 <NA> 5
## 9 2007-09-09 55 50 52 88 Rain 7
## 10 2007-09-10 59 50 54 82 Rain 4
## # ... with 3,651 more rows# Plot to check work
ggplot(akl_daily, aes(x = date, y = max_temp)) +
geom_line() ## Warning: Removed 1 rows containing missing values (geom_path).
# Import "akl_weather_hourly_2016.csv"
akl_hourly_raw <- read_csv("./RInputFiles/akl_weather_hourly_2016.csv")## Parsed with column specification:
## cols(
## year = col_integer(),
## month = col_integer(),
## mday = col_integer(),
## time = col_time(format = ""),
## temperature = col_double(),
## weather = col_character(),
## conditions = col_character(),
## events = col_character(),
## humidity = col_integer(),
## date_utc = col_datetime(format = "")
## )# Print akl_hourly_raw
akl_hourly_raw## # A tibble: 17,454 x 10
## year month mday time temperature weather conditions events humidity
## <int> <int> <int> <time> <dbl> <chr> <chr> <chr> <int>
## 1 2016 1 1 00:00 68.0 Clear Clear <NA> 68
## 2 2016 1 1 00:30 68.0 Clear Clear <NA> 68
## 3 2016 1 1 01:00 68.0 Clear Clear <NA> 73
## 4 2016 1 1 01:30 68.0 Clear Clear <NA> 68
## 5 2016 1 1 02:00 68.0 Clear Clear <NA> 68
## 6 2016 1 1 02:30 68.0 Clear Clear <NA> 68
## 7 2016 1 1 03:00 68.0 Clear Clear <NA> 68
## 8 2016 1 1 03:30 68.0 Cloudy Partly Cl~ <NA> 68
## 9 2016 1 1 04:00 68.0 Cloudy Scattered~ <NA> 68
## 10 2016 1 1 04:30 66.2 Cloudy Partly Cl~ <NA> 73
## # ... with 17,444 more rows, and 1 more variable: date_utc <dttm># Use make_date() to combine year, month and mday
akl_hourly <- akl_hourly_raw %>%
mutate(date = make_date(year = year, month = month, day = mday))
# Parse datetime_string
akl_hourly <- akl_hourly %>%
mutate(
datetime_string = paste(date, time, sep = "T"),
datetime = ymd_hms(datetime_string)
)
# Print date, time and datetime columns of akl_hourly
akl_hourly %>% select(date, time, datetime)## # A tibble: 17,454 x 3
## date time datetime
## <date> <time> <dttm>
## 1 2016-01-01 00:00 2016-01-01 00:00:00
## 2 2016-01-01 00:30 2016-01-01 00:30:00
## 3 2016-01-01 01:00 2016-01-01 01:00:00
## 4 2016-01-01 01:30 2016-01-01 01:30:00
## 5 2016-01-01 02:00 2016-01-01 02:00:00
## 6 2016-01-01 02:30 2016-01-01 02:30:00
## 7 2016-01-01 03:00 2016-01-01 03:00:00
## 8 2016-01-01 03:30 2016-01-01 03:30:00
## 9 2016-01-01 04:00 2016-01-01 04:00:00
## 10 2016-01-01 04:30 2016-01-01 04:30:00
## # ... with 17,444 more rows# Plot to check work
ggplot(akl_hourly, aes(x = datetime, y = temperature)) +
geom_line()
# Examine the head() of release_time
releases <- read_csv("./RInputFiles/rversions.csv")## Parsed with column specification:
## cols(
## major = col_integer(),
## minor = col_integer(),
## patch = col_integer(),
## date = col_date(format = ""),
## datetime = col_datetime(format = ""),
## time = col_time(format = ""),
## type = col_character()
## )release_time <- releases %>% pull(datetime)
head(release_time)## [1] "1997-12-04 08:47:58 UTC" "1997-12-21 13:09:22 UTC"
## [3] "1998-01-10 00:31:55 UTC" "1998-03-14 19:25:55 UTC"
## [5] "1998-05-02 07:58:17 UTC" "1998-06-14 12:56:20 UTC"# Examine the head() of the months of release_time
head(month(release_time))## [1] 12 12 1 3 5 6# Extract the month of releases
month(release_time) %>% table()## .
## 1 2 3 4 5 6 7 8 9 10 11 12
## 5 6 8 18 5 16 4 7 2 15 6 13# Extract the year of releases
year(release_time) %>% table()## .
## 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
## 2 10 9 6 6 5 5 4 4 4 4 6 5 4 6
## 2012 2013 2014 2015 2016 2017
## 4 4 4 5 5 3# How often is the hour before 12 (noon)?
mean(hour(release_time) < 12)## [1] 0.752381# How often is the release in am?
mean(am(release_time))## [1] 0.752381# Use wday() to tabulate release by day of the week
wday(releases$datetime) %>% table()## .
## 1 2 3 4 5 6 7
## 3 29 9 12 18 31 3# Add label = TRUE to make table more readable
wday(releases$datetime, label=TRUE) %>% table()## .
## Sun Mon Tue Wed Thu Fri Sat
## 3 29 9 12 18 31 3# Create column wday to hold labelled week days
releases$wday <- wday(releases$datetime, label=TRUE)
# Plot barchart of weekday by type of release
ggplot(releases, aes(x=wday)) +
geom_bar() +
facet_wrap(~ type, ncol = 1, scale = "free_y")
# Add columns for year, yday and month
akl_daily <- akl_daily %>%
mutate(
year = year(date),
yday = yday(date),
month = month(date, label=TRUE))
# Plot max_temp by yday for all years
ggplot(akl_daily, aes(x = yday, y = max_temp)) +
geom_line(aes(group = year), alpha = 0.5)## Warning: Removed 1 rows containing missing values (geom_path).
# Examine distribtion of max_temp by month
ggplot(akl_daily, aes(x = max_temp, y = month, height = ..density..)) +
geom_density_ridges(stat = "density")## Warning: Removed 10 rows containing non-finite values (stat_density).
# Create new columns hour, month and rainy
akl_hourly <- akl_hourly %>%
mutate(
hour = hour(datetime),
month = month(datetime, label=TRUE),
rainy = (weather == "Precipitation")
)
# Filter for hours between 8am and 10pm (inclusive)
akl_day <- akl_hourly %>%
filter(hour >= 8, hour <= 22)
# Summarise for each date if there is any rain
rainy_days <- akl_day %>%
group_by(month, date) %>%
summarise(
any_rain = any(rainy)
)
# Summarise for each month, the number of days with rain
rainy_days %>%
summarise(
days_rainy = sum(any_rain)
)## # A tibble: 12 x 2
## month days_rainy
## <ord> <int>
## 1 Jan 15
## 2 Feb 13
## 3 Mar 12
## 4 Apr 15
## 5 May 21
## 6 Jun 19
## 7 Jul 22
## 8 Aug 16
## 9 Sep 25
## 10 Oct 20
## 11 Nov 19
## 12 Dec 11r_3_4_1 <- ymd_hms("2016-05-03 07:13:28 UTC")
# Round down to day
floor_date(r_3_4_1, unit = "day")## [1] "2016-05-03 UTC"# Round to nearest 5 minutes
round_date(r_3_4_1, unit = "5 minutes")## [1] "2016-05-03 07:15:00 UTC"# Round up to week
ceiling_date(r_3_4_1, unit = "week")## [1] "2016-05-08 UTC"# Subtract r_3_4_1 rounded down to day
r_3_4_1 - floor_date(r_3_4_1, unit = "day")## Time difference of 7.224444 hours# Create day_hour, datetime rounded down to hour
akl_hourly <- akl_hourly %>%
mutate(
day_hour = floor_date(datetime, unit = "hour")
)
# Count observations per hour
akl_hourly %>%
count(day_hour) ## # A tibble: 8,770 x 2
## day_hour n
## <dttm> <int>
## 1 2016-01-01 00:00:00 2
## 2 2016-01-01 01:00:00 2
## 3 2016-01-01 02:00:00 2
## 4 2016-01-01 03:00:00 2
## 5 2016-01-01 04:00:00 2
## 6 2016-01-01 05:00:00 2
## 7 2016-01-01 06:00:00 2
## 8 2016-01-01 07:00:00 2
## 9 2016-01-01 08:00:00 2
## 10 2016-01-01 09:00:00 2
## # ... with 8,760 more rows# Find day_hours with n != 2
akl_hourly %>%
count(day_hour) %>%
filter(n != 2) %>%
arrange(desc(n))## # A tibble: 92 x 2
## day_hour n
## <dttm> <int>
## 1 2016-04-03 02:00:00 4
## 2 2016-09-25 00:00:00 4
## 3 2016-06-26 09:00:00 1
## 4 2016-09-01 23:00:00 1
## 5 2016-09-02 01:00:00 1
## 6 2016-09-04 11:00:00 1
## 7 2016-09-04 16:00:00 1
## 8 2016-09-04 17:00:00 1
## 9 2016-09-05 00:00:00 1
## 10 2016-09-05 15:00:00 1
## # ... with 82 more rowsChapter 3 - Arithmetic with Dates and Times
Taking differences of datetimes:
- Pure subtraction will give the days between two datetimes, reported on the command line as “Time difference of x days”
- The difftime(day1, day2, units=) function is the same as day1 - day2, but with additional control of being able to request units (secs, mins, hours, days, weeks)
- The difftime(day1, day2, units=) function is the same as day1 - day2, but with additional control of being able to request units (secs, mins, hours, days, weeks)
- The today() function gives you today’s date as a Date object
- The now() function gives you the current date-time as a POSIXct object
Time spans - difficult because they do not have a constant meaning (e.g., impact of daylight savings time):
- The lubridate package manages time spans as EITHER period or duration
- The period is the way a human thinks about it - 1 day means same exact hour-minute-second tomorrow
- The duration is the way a stopwatch thinks about it - 1 day means 24 hours from now
- The period is the way a human thinks about it - 1 day means same exact hour-minute-second tomorrow
- The period time span in lubridate is called by adding an “s” to the end of the relevant function
- For example, days(x=1) will be exactly +1 in the days category only (all other units untouched)
- For example, days(x=1) will be exactly +1 in the days category only (all other units untouched)
- The duration in lubridate is called by adding a “d” to the front of the relevant period function
- For example, ddays(x=1) will add 24 hours to the datetime
- For example, ddays(x=1) will add 24 hours to the datetime
- There was an eclipse over North America on 2017-08-21 at 18:26:40
- It’s possible to predict the next eclipse with similar geometry by calculating the time and date one Saros in the future
- A Saros is a length of time that corresponds to 223 Synodic months, a Synodic month being the period of the Moon’s phases, a duration of 29 days, 12 hours, 44 minutes and 3 seconds
- It’s possible to predict the next eclipse with similar geometry by calculating the time and date one Saros in the future
- What should ymd(“2018-01-31”) + months(1) return? Should it be 30, 31 or 28 days in the future? Try it
- In general lubridate returns the same day of the month in the next month, but since the 31st of February doesn’t exist lubridate returns a missing value, NA
- There are alternative addition and subtraction operators: %m+% and %m-% that have different behavior
- Rather than returning an NA for a non-existent date, they roll back to the last existing date
- But use these operators with caution, unlike + and -, you might not get x back from x %m+% months(1) %m-% months(1)
- If you’d prefer that the date was rolled forward check out add_with_rollback() which has roll_to_first argument
- In general lubridate returns the same day of the month in the next month, but since the 31st of February doesn’t exist lubridate returns a missing value, NA
Intervals - third option in lubridate for storing times:
- Can find length, whether an object is in the interval, whether various intervals overlap, and the like
- Intervals can be created either by using interval(datetime1, datetime2) or datetime1 %–% datetime2
- Intervals can be created either by using interval(datetime1, datetime2) or datetime1 %–% datetime2
- There are many lubridate functions for working with intervals
- int_start() and int_end() will give back the start and end date for the interval
- int_length() will give back the interval length in seconds
- as.period() will return the interval length as a period, while as.duration() will return the interval length as a duration
- aDateTime %within% anInterval will return a boolean that answers the question
- The int_overlaps(int1, int2) will return a boolean for whether there is any overlap
- int_start() and int_end() will give back the start and end date for the interval
- Intervals tend to be best when you have a specific start and end date
- Otherwise, use periods for human purposes and durations for technical purposes
- Otherwise, use periods for human purposes and durations for technical purposes
- The operator %within% tests if the datetime (or interval) on the left hand side is within the interval of the right hand side
- int_overlaps() performs a similar test, but will return true if two intervals overlap at all
Example code includes:
# The date of landing and moment of step
date_landing <- mdy("July 20, 1969")
moment_step <- mdy_hms("July 20, 1969, 02:56:15", tz = "UTC")
# How many days since the first man on the moon?
difftime(today(), date_landing, units = "days")## Time difference of 17792 days# How many seconds since the first man on the moon?
difftime(now(), moment_step, units = "secs")## Time difference of 1537270659 secs# Three dates
mar_11 <- ymd_hms("2017-03-11 12:00:00",
tz = "America/Los_Angeles")
mar_12 <- ymd_hms("2017-03-12 12:00:00",
tz = "America/Los_Angeles")
mar_13 <- ymd_hms("2017-03-13 12:00:00",
tz = "America/Los_Angeles")
# Difference between mar_13 and mar_12 in seconds
difftime(mar_13, mar_12, units = "secs")## Time difference of 86400 secs# Difference between mar_12 and mar_11 in seconds
difftime(mar_12, mar_11, units = "secs")## Time difference of 82800 secs# Add a period of one week to mon_2pm
mon_2pm <- dmy_hm("27 Aug 2018 14:00")
mon_2pm + weeks(1)## [1] "2018-09-03 14:00:00 UTC"# Add a duration of 81 hours to tue_9am
tue_9am <- dmy_hm("28 Aug 2018 9:00")
tue_9am + dhours(81)## [1] "2018-08-31 18:00:00 UTC"# Subtract a period of five years from today()
today() - years(5)## [1] "2013-04-06"# Subtract a duration of five years from today()
today() - dyears(5)## [1] "2013-04-07"# Time of North American Eclipse 2017
eclipse_2017 <- ymd_hms("2017-08-21 18:26:40")
# Duration of 29 days, 12 hours, 44 mins and 3 secs
synodic <- ddays(29) + dhours(12) + dminutes(44) + dseconds(3)
# 223 synodic months
saros <- 223 * synodic
# Add saros to eclipse_2017
eclipse_2017 + saros## [1] "2035-09-02 02:09:49 UTC"# Add a period of 8 hours to today
today_8am <- today() + hours(8)
# Sequence of two weeks from 1 to 26
every_two_weeks <- 1:26 * weeks(2)
# Create datetime for every two weeks for a year
today_8am + every_two_weeks## [1] "2018-04-20 08:00:00 UTC" "2018-05-04 08:00:00 UTC"
## [3] "2018-05-18 08:00:00 UTC" "2018-06-01 08:00:00 UTC"
## [5] "2018-06-15 08:00:00 UTC" "2018-06-29 08:00:00 UTC"
## [7] "2018-07-13 08:00:00 UTC" "2018-07-27 08:00:00 UTC"
## [9] "2018-08-10 08:00:00 UTC" "2018-08-24 08:00:00 UTC"
## [11] "2018-09-07 08:00:00 UTC" "2018-09-21 08:00:00 UTC"
## [13] "2018-10-05 08:00:00 UTC" "2018-10-19 08:00:00 UTC"
## [15] "2018-11-02 08:00:00 UTC" "2018-11-16 08:00:00 UTC"
## [17] "2018-11-30 08:00:00 UTC" "2018-12-14 08:00:00 UTC"
## [19] "2018-12-28 08:00:00 UTC" "2019-01-11 08:00:00 UTC"
## [21] "2019-01-25 08:00:00 UTC" "2019-02-08 08:00:00 UTC"
## [23] "2019-02-22 08:00:00 UTC" "2019-03-08 08:00:00 UTC"
## [25] "2019-03-22 08:00:00 UTC" "2019-04-05 08:00:00 UTC"jan_31 <- ymd("2018-01-31")
# A sequence of 1 to 12 periods of 1 month
month_seq <- 1:12 * months(1)
# Add 1 to 12 months to jan_31
jan_31 + month_seq## [1] NA "2018-03-31" NA "2018-05-31" NA
## [6] "2018-07-31" "2018-08-31" NA "2018-10-31" NA
## [11] "2018-12-31" "2019-01-31"# Replace + with %m+%
jan_31 %m+% month_seq## [1] "2018-02-28" "2018-03-31" "2018-04-30" "2018-05-31" "2018-06-30"
## [6] "2018-07-31" "2018-08-31" "2018-09-30" "2018-10-31" "2018-11-30"
## [11] "2018-12-31" "2019-01-31"# Replace + with %m-%
jan_31 %m-% month_seq## [1] "2017-12-31" "2017-11-30" "2017-10-31" "2017-09-30" "2017-08-31"
## [6] "2017-07-31" "2017-06-30" "2017-05-31" "2017-04-30" "2017-03-31"
## [11] "2017-02-28" "2017-01-31"# Create monarchs
mNames <- c('Elizabeth II' ,'Victoria' ,'George V' ,'George III' ,'George VI' ,'George IV' ,'Edward VII' ,'William IV' ,'Edward VIII' ,'George III(also United Kingdom)' ,'George II' ,'George I' ,'Anne' ,'Henry III' ,'Edward III' ,'Elizabeth I' ,'Henry VI' ,'Henry VI' ,'Æthelred II' ,'Æthelred II' ,'Henry VIII' ,'Charles II' ,'Henry I' ,'Henry II(co-ruler with Henry the Young King)' ,'Edward I' ,'Alfred the Great' ,'Edward the Elder' ,'Charles I' ,'Henry VII' ,'Edward the Confessor' ,'Richard II' ,'James I' ,'Edward IV' ,'Edward IV' ,'William I' ,'Edward II' ,'Cnut' ,'Stephen' ,'Stephen' ,'John' ,'Edgar I' ,'Æthelstan' ,'Henry IV' ,'William III(co-ruler with Mary II)' ,'Henry the Young King(co-ruler with Henry II)' ,'William II' ,'Richard I' ,'Eadred' ,'Henry V' ,'Edmund I' ,'Edward VI' ,'Mary II(co-ruler with William III)' ,'Mary I' ,'Anne(also Kingdom of Great Britain)' ,'Eadwig' ,'James II' ,'Edward the Martyr' ,'Harold I' ,'Harthacnut' ,'Richard III' ,'Louis (disputed)' ,'Harold II' ,'Edmund II' ,'Matilda (disputed)' ,'Edward V' ,'Edgar II' ,'Sweyn Forkbeard' ,'Jane (disputed)' ,'James VI' ,'William I' ,'Constantine II' ,'David II' ,'Alexander III' ,'Malcolm III' ,'Alexander II' ,'James I' ,'Malcolm II' ,'James V' ,'David I' ,'James III' ,'Charles II' ,'Charles II' ,'James IV' ,'Mary I' ,'Charles I' ,'Kenneth II' ,'James II' ,'Robert I' ,'Robert II' ,'Alexander I' ,'Macbeth' ,'Robert III' ,'Constantine I' ,'Kenneth MacAlpin' ,'William II' ,'Malcolm IV' ,'Giric(co-ruler with Eochaid?)' ,'Donald II' ,'Malcolm I' ,'Edgar' ,'Kenneth III' ,'Indulf' ,'Duncan I' ,'Mary II' ,'Amlaíb' ,'Anne(also Kingdom of Great Britain)' ,'Dub' ,'Cuilén' ,'Domnall mac Ailpín' ,'James VII' ,'Margaret' ,'John Balliol' ,'Donald III' ,'Constantine III' ,'Áed mac Cináeda' ,'Lulach' ,'Duncan II' ,'Ruaidrí Ua Conchobair' ,'Edward Bruce (disputed)' ,'Brian Ua Néill (disputed)' ,'Gruffudd ap Cynan' ,'Llywelyn the Great' ,'Owain Gwynedd' ,'Dafydd ab Owain Gwynedd' ,'Hywel ab Owain Gwynedd' ,'Llywelyn ap Gruffudd' ,'Owain Glyndŵr (disputed)' ,'Owain Goch ap Gruffydd' ,'Owain Lawgoch (disputed)' ,'Dafydd ap Llywelyn' ,'Dafydd ap Gruffydd')
mDominion <- c('United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'United Kingdom' ,'Great Britain' ,'Great Britain' ,'Great Britain' ,'Great Britain' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'England' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Scotland' ,'Ireland' ,'Ireland' ,'Ireland' ,'Gwynedd' ,'Gwynedd' ,'Gwynedd' ,'Gwynedd' ,'Gwynedd' ,'Wales' ,'Wales' ,'Wales' ,'Wales' ,'Wales' ,'Wales')
mFrom <- c('1952-02-06' ,'1837-06-20' ,'1910-05-06' ,'1801-01-01' ,'1936-12-11' ,'1820-01-29' ,'1901-01-22' ,'1830-06-26' ,'1936-01-20' ,'1760-10-25' ,'1727-06-22' ,'1714-08-01' ,'1707-05-01' ,'NA' ,'1327-01-25' ,'1558-11-17' ,'1422-08-31' ,'1470-10-31' ,'978-03-18' ,'1014-02-03' ,'1509-04-22' ,'1649-01-30' ,'1100-08-03' ,'1154-10-25' ,'1272-11-20' ,'871-04-24' ,'899-10-27' ,'1625-03-27' ,'1485-08-22' ,'1042-06-08' ,'1377-06-22' ,'1603-03-24' ,'1461-03-04' ,'1471-04-11' ,'1066-12-12' ,'1307-07-07' ,'1016-11-30' ,'1135-12-22' ,'1141-11-01' ,'1199-04-06' ,'959-10-01' ,'924-08-02' ,'1399-09-29' ,'1689-02-13' ,'1170-06-14' ,'1087-09-09' ,'1189-07-06' ,'946-05-26' ,'1413-03-21' ,'939-10-27' ,'1547-01-28' ,'1689-02-13' ,'1553-07-19' ,'1702-03-08' ,'955-11-23' ,'1685-02-06' ,'975-07-09' ,'1037-11-12' ,'1040-03-17' ,'1483-06-26' ,'1216-06-14' ,'1066-01-05' ,'1016-04-23' ,'1141-04-07' ,'1483-04-09' ,'1066-10-15' ,'1013-12-25' ,'1553-07-10' ,'1567-07-24' ,'1165-12-09' ,'900-01-01' ,'1329-06-07' ,'1249-07-06' ,'1058-03-17' ,'1214-12-04' ,'1406-04-04' ,'1005-03-25' ,'1513-09-09' ,'1124-04-23' ,'1460-08-03' ,'1649-01-30' ,'1660-05-29' ,'1488-06-11' ,'1542-12-14' ,'1625-03-27' ,'971-01-01' ,'1437-02-21' ,'1306-03-25' ,'1371-02-22' ,'1107-01-08' ,'1040-08-14' ,'1390-04-19' ,'862-01-01' ,'843-01-01' ,'1689-05-11' ,'1153-05-24' ,'878-01-01' ,'889-01-01' ,'943-01-01' ,'1097-01-01' ,'997-01-01' ,'954-01-01' ,'1034-11-25' ,'1689-04-11' ,'971-01-01' ,'1702-03-08' ,'962-01-01' ,'NA' ,'858-01-01' ,'1685-02-06' ,'1286-11-25' ,'1292-11-17' ,'1093-11-13' ,'1095-01-01' ,'877-01-01' ,'1057-08-15' ,'1094-05-01' ,'1166-01-01' ,'1315-06-01' ,'1258-01-01' ,'1081-01-01' ,'1195-01-01' ,'1137-01-01' ,'1170-01-01' ,'1170-01-01' ,'1253-01-01' ,'1400-09-16' ,'1246-02-25' ,'1372-05-01' ,'1240-04-12' ,'1282-12-11')
mTo <- c('2018-02-08' ,'1901-01-22' ,'1936-01-20' ,'1820-01-29' ,'1952-02-06' ,'1830-06-26' ,'1910-05-06' ,'1837-06-20' ,'1936-12-11' ,'1801-01-01' ,'1760-10-25' ,'1727-06-11' ,'1714-08-01' ,'1272-11-16' ,'1377-06-21' ,'1603-03-24' ,'1461-03-04' ,'1471-04-11' ,'1013-12-25' ,'1016-04-23' ,'1547-01-28' ,'1685-02-06' ,'1135-12-01' ,'1189-07-06' ,'1307-07-07' ,'899-10-26' ,'924-07-17' ,'1649-01-30' ,'1509-04-21' ,'1066-01-05' ,'1399-09-29' ,'1625-03-27' ,'1470-10-03' ,'1483-04-09' ,'1087-09-09' ,'1327-01-20' ,'1035-11-12' ,'1141-04-07' ,'1154-10-25' ,'1216-10-19' ,'975-07-08' ,'939-10-27' ,'1413-03-20' ,'1702-03-08' ,'1183-06-11' ,'1100-08-02' ,'1199-04-06' ,'955-11-23' ,'1422-08-31' ,'946-05-26' ,'1553-07-06' ,'1694-12-28' ,'1558-11-17' ,'1707-04-30' ,'959-10-01' ,'1688-12-11' ,'978-03-18' ,'1040-03-17' ,'1042-06-08' ,'1485-08-22' ,'1217-09-22' ,'1066-10-14' ,'1016-11-30' ,'1141-11-01' ,'1483-06-26' ,'1066-12-17' ,'1014-02-03' ,'1553-07-19' ,'1625-03-27' ,'1214-12-04' ,'943-01-01' ,'1371-02-22' ,'1286-03-19' ,'1093-11-13' ,'1249-07-06' ,'1437-02-21' ,'1034-11-25' ,'1542-12-14' ,'1153-05-24' ,'1488-06-11' ,'1651-09-03' ,'1685-02-06' ,'1513-09-09' ,'1567-07-24' ,'1649-01-30' ,'995-01-01' ,'1460-08-03' ,'1329-06-07' ,'1390-04-19' ,'1124-04-23' ,'1057-08-15' ,'1406-04-04' ,'877-01-01' ,'858-02-13' ,'1702-03-08' ,'1165-12-09' ,'889-01-01' ,'900-01-01' ,'954-01-01' ,'1107-01-08' ,'1005-03-25' ,'962-01-01' ,'1040-08-14' ,'1694-12-28' ,'977-01-01' ,'1707-04-30' ,'NA' ,'971-01-01' ,'862-04-13' ,'1688-12-11' ,'1290-09-26' ,'1296-07-10' ,'1097-01-01' ,'1097-01-01' ,'878-01-01' ,'1058-03-17' ,'1094-11-12' ,'1193-01-01' ,'1318-10-14' ,'1260-01-01' ,'1137-01-01' ,'1240-04-11' ,'1170-01-01' ,'1195-01-01' ,'1170-01-01' ,'1282-12-11' ,'1416-01-01' ,'1255-01-01' ,'1378-07-01' ,'1246-02-25' ,'1283-10-03')
padMDate <- function(x) {
if (is.na(x[1]) | x[1] == "NA") {
NA
} else {
paste0(c(str_pad(x[1], 4, pad="0"), x[2], x[3]), collapse="-")
}
}
monarchs <- tibble::tibble(name=mNames, dominion=mDominion,
from=ymd(sapply(str_split(mFrom, "-"), FUN=padMDate)),
to=ymd(sapply(str_split(mTo, "-"), FUN=padMDate))
)
# Print monarchs
monarchs## # A tibble: 131 x 4
## name dominion from to
## <chr> <chr> <date> <date>
## 1 Elizabeth II United Kingdom 1952-02-06 2018-02-08
## 2 Victoria United Kingdom 1837-06-20 1901-01-22
## 3 George V United Kingdom 1910-05-06 1936-01-20
## 4 George III United Kingdom 1801-01-01 1820-01-29
## 5 George VI United Kingdom 1936-12-11 1952-02-06
## 6 George IV United Kingdom 1820-01-29 1830-06-26
## 7 Edward VII United Kingdom 1901-01-22 1910-05-06
## 8 William IV United Kingdom 1830-06-26 1837-06-20
## 9 Edward VIII United Kingdom 1936-01-20 1936-12-11
## 10 George III(also United Kingdom) Great Britain 1760-10-25 1801-01-01
## # ... with 121 more rows# Create an interval for reign
monarchs <- monarchs %>%
mutate(reign = from %--% to)
# Find the length of reign, and arrange
monarchs %>%
mutate(length = int_length(reign)) %>%
arrange(desc(length)) %>%
select(name, length, dominion)## # A tibble: 131 x 3
## name length dominion
## <chr> <dbl> <chr>
## 1 Elizabeth II 2083017600 United Kingdom
## 2 Victoria 2006726400 United Kingdom
## 3 James VI 1820102400 Scotland
## 4 Gruffudd ap Cynan 1767139200 Gwynedd
## 5 Edward III 1590624000 England
## 6 William I 1545868800 Scotland
## 7 Llywelyn the Great 1428796800 Gwynedd
## 8 Elizabeth I 1399507200 England
## 9 Constantine II 1356912000 Scotland
## 10 David II 1316304000 Scotland
## # ... with 121 more rows# Print halleys
pDate <- c('66-01-26', '141-03-25', '218-04-06', '295-04-07', '374-02-13', '451-07-03', '530-11-15', '607-03-26', '684-11-26', '760-06-10', '837-02-25', '912-07-27', '989-09-02', '1066-03-25', '1145-04-19', '1222-09-10', '1301-10-22', '1378-11-09', '1456-01-08', '1531-08-26', '1607-10-27', '1682-09-15', '1758-03-13', '1835-11-16', '1910-04-20', '1986-02-09', '2061-07-28')
sDate <- c('66-01-25', '141-03-22', '218-04-06', '295-04-07', '374-02-13', '451-06-28', '530-09-27', '607-03-15', '684-10-02', '760-05-20', '837-02-25', '912-07-18', '989-09-02', '1066-01-01', '1145-04-15', '1222-09-10', '1301-10-22', '1378-11-09', '1456-01-08', '1531-08-26', '1607-10-27', '1682-09-15', '1758-03-13', '1835-08-01', '1910-04-20', '1986-02-09', '2061-07-28')
eDate <- c('66-01-26', '141-03-25', '218-05-17', '295-04-20', '374-02-16', '451-07-03', '530-11-15', '607-03-26', '684-11-26', '760-06-10', '837-02-28', '912-07-27', '989-09-05', '1066-03-25', '1145-04-19', '1222-09-28', '1301-10-31', '1378-11-14', '1456-06-09', '1531-08-26', '1607-10-27', '1682-09-15', '1758-12-25', '1835-11-16', '1910-05-20', '1986-02-09', '2061-07-28')
halleys <- tibble::tibble(perihelion_date=ymd(sapply(str_split(pDate, "-"), FUN=padMDate)),
start_date=ymd(sapply(str_split(sDate, "-"), FUN=padMDate)),
end_date=ymd(sapply(str_split(eDate, "-"), FUN=padMDate))
)
# New column for interval from start to end date
halleys <- halleys %>%
mutate(visible = start_date %--% end_date)
# The visitation of 1066
halleys_1066 <- halleys[14, ]
# Monarchs in power on perihelion date
monarchs %>%
filter(halleys_1066$perihelion_date %within% reign) %>%
select(name, from, to, dominion)## # A tibble: 2 x 4
## name from to dominion
## <chr> <date> <date> <chr>
## 1 Harold II 1066-01-05 1066-10-14 England
## 2 Malcolm III 1058-03-17 1093-11-13 Scotland# Monarchs whose reign overlaps visible time
monarchs %>%
filter(int_overlaps(halleys_1066$visible, reign)) %>%
select(name, from, to, dominion)## # A tibble: 3 x 4
## name from to dominion
## <chr> <date> <date> <chr>
## 1 Edward the Confessor 1042-06-08 1066-01-05 England
## 2 Harold II 1066-01-05 1066-10-14 England
## 3 Malcolm III 1058-03-17 1093-11-13 Scotland# New columns for duration and period
monarchs <- monarchs %>%
mutate(
duration = as.duration(reign),
period = as.period(reign))
# Examine results
monarchs %>%
select(name, duration, period) %>%
head(10) %>%
print.data.frame()## name duration
## 1 Elizabeth II 2083017600s (~66.01 years)
## 2 Victoria 2006726400s (~63.59 years)
## 3 George V 811296000s (~25.71 years)
## 4 George III 601948800s (~19.07 years)
## 5 George VI 478224000s (~15.15 years)
## 6 George IV 328406400s (~10.41 years)
## 7 Edward VII 292982400s (~9.28 years)
## 8 William IV 220406400s (~6.98 years)
## 9 Edward VIII 28166400s (~46.57 weeks)
## 10 George III(also United Kingdom) 1268092800s (~40.18 years)
## period
## 1 66y 0m 2d 0H 0M 0S
## 2 63y 7m 2d 0H 0M 0S
## 3 25y 8m 14d 0H 0M 0S
## 4 19y 0m 28d 0H 0M 0S
## 5 15y 1m 26d 0H 0M 0S
## 6 10y 4m 28d 0H 0M 0S
## 7 9y 3m 14d 0H 0M 0S
## 8 6y 11m 25d 0H 0M 0S
## 9 10m 21d 0H 0M 0S
## 10 40y 2m 7d 0H 0M 0SChapter 4 - Problems in Practice
Time zones - ways to keep track of times in different locations (can pose analysis challenges):
- Typically captured as an offset from GMT, but specified in R using tz= since the offset to GMT can change during the year (DST for example)
- Sys.timezone() gives the timezone on your computer
- OlsonNames() gives all the timezones that R is aware of
- The OlsonNames() function matches with an international standard as to which cities are included
- The lubridate::tz() will extract the timezone from a specific datetime
- Sys.timezone() gives the timezone on your computer
- Can change the timezone without changing the underlying clock time components by using lubridate::force_tz()
- force_tz(ymd_hm(“2017-12-12 12:00”, tz=“America/Los_Angeles”), tzone=“America/Boston”) will produce 2017-12-12 12:00 EST (note that the 12:00 is held, with ONLY time-zone changed)
- force_tz(ymd_hm(“2017-12-12 12:00”, tz=“America/Los_Angeles”), tzone=“America/Boston”) will produce 2017-12-12 12:00 EST (note that the 12:00 is held, with ONLY time-zone changed)
- Can view the time in a different zone by using lubridate::with_tz()
- with_tz(ymd_hm(“2017-12-12 12:00”, tz=“America/Los_Angeles”), tzone=“America/Boston”) will produce 2017-12-12 15:00 EST (note that 15:00 EST and 12:00 PST are the same)
- with_tz(ymd_hm(“2017-12-12 12:00”, tz=“America/Los_Angeles”), tzone=“America/Boston”) will produce 2017-12-12 15:00 EST (note that 15:00 EST and 12:00 PST are the same)
- For this entire course, if you’ve ever had a time, it’s always had an accompanying date, i.e. a datetime. But sometimes you just have a time without a date
- If you find yourself in this situation, the hms package provides an hms class of object for holding times without dates, and the best place to start would be with as.hms()
- readr knows the hms class, so if it comes across something that looks like a time it will use it
- If you find yourself in this situation, the hms package provides an hms class of object for holding times without dates, and the best place to start would be with as.hms()
Importing and exporting datetimes:
- The parse_date_time() function is designed to be forgiving and flexible, but at the expense of being slow (since it considers many possible formats)
- The fasttime::fastPOSIXct() is designed to very quickly read a proper ISO formatting of “YYYY-MM-DD”
- The lubridate::fast_strptime(x=, format=) is also fast, but it requires a valid strptime format like “%Y-%m-%d” rather than the more flexible/forgiving parse_date_time(x=, order=“ymd”)
- See help for strptime() for the valid strings
- The fasttime::fastPOSIXct() is designed to very quickly read a proper ISO formatting of “YYYY-MM-DD”
- The readr::write_csv() will write datetime objects in a proper ISO format, making for easy read-in
- Can also use the lubridate::stamp() capability to build a function that will format things based on an example you provide
- my_stamp <- stamp(“Tuesday October 10 2017”)
- my_stamp has been created by lubridate::stamp() as function(x) format(x, format=“%A %B %d %Y”) to match the example given
- my_stamp <- stamp(“Tuesday October 10 2017”)
Wrap-up:
- Chapter 1: base R objects Date, POSIXct
- lubridate, zoo, xts, and the like all work together with each other and these
- lubridate, zoo, xts, and the like all work together with each other and these
- Chapter 2: importing and manipulating datetime obects
- Chapter 3: challenges of arithmetic with datetimes
- periods, durations, intervals
- periods, durations, intervals
- Chapter 4: time zones, and import/outputs
Example code includes:
# Game2: CAN vs NZL in Edmonton
game2 <- mdy_hm("June 11 2015 19:00")
# Game3: CHN vs NZL in Winnipeg
game3 <- mdy_hm("June 15 2015 18:30")
# Set the timezone to "America/Edmonton"
game2_local <- force_tz(game2, tzone = "America/Edmonton")
game2_local## [1] "2015-06-11 19:00:00 MDT"# Set the timezone to "America/Winnipeg"
game3_local <- force_tz(game3, tzone = "America/Winnipeg")
game3_local## [1] "2015-06-15 18:30:00 CDT"# How long does the team have to rest?
as.period(game2_local %--% game3_local)## [1] "3d 22H 30M 0S"# What time is game2_local in NZ?
with_tz(game2_local, tzone = "Pacific/Auckland")## [1] "2015-06-12 13:00:00 NZST"# What time is game2_local in Corvallis, Oregon?
with_tz(game2_local, tzone = "America/Los_Angeles")## [1] "2015-06-11 18:00:00 PDT"# What time is game3_local in NZ?
with_tz(game3_local, tzone = "Pacific/Auckland")## [1] "2015-06-16 11:30:00 NZST"# Examine datetime and date_utc columns
head(akl_hourly$datetime)## [1] "2016-01-01 00:00:00 UTC" "2016-01-01 00:30:00 UTC"
## [3] "2016-01-01 01:00:00 UTC" "2016-01-01 01:30:00 UTC"
## [5] "2016-01-01 02:00:00 UTC" "2016-01-01 02:30:00 UTC"head(akl_hourly$date_utc)## [1] "2015-12-31 11:00:00 UTC" "2015-12-31 11:30:00 UTC"
## [3] "2015-12-31 12:00:00 UTC" "2015-12-31 12:30:00 UTC"
## [5] "2015-12-31 13:00:00 UTC" "2015-12-31 13:30:00 UTC"# Force datetime to Pacific/Auckland
akl_hourly <- akl_hourly %>%
mutate(
datetime = force_tz(datetime, tzone = "Pacific/Auckland"))
# Reexamine datetime
head(akl_hourly$datetime)## [1] "2016-01-01 00:00:00 NZDT" "2016-01-01 00:30:00 NZDT"
## [3] "2016-01-01 01:00:00 NZDT" "2016-01-01 01:30:00 NZDT"
## [5] "2016-01-01 02:00:00 NZDT" "2016-01-01 02:30:00 NZDT"# Are datetime and date_utc the same moments
table(akl_hourly$datetime - akl_hourly$date_utc)##
## -82800 0 3600
## 2 17450 2# Import auckland hourly data
akl_hourly <- read_csv("./RInputFiles/akl_weather_hourly_2016.csv")## Parsed with column specification:
## cols(
## year = col_integer(),
## month = col_integer(),
## mday = col_integer(),
## time = col_time(format = ""),
## temperature = col_double(),
## weather = col_character(),
## conditions = col_character(),
## events = col_character(),
## humidity = col_integer(),
## date_utc = col_datetime(format = "")
## )# Examine structure of time column
str(akl_hourly$time)## Classes 'hms', 'difftime' atomic [1:17454] 0 1800 3600 5400 7200 9000 10800 12600 14400 16200 ...
## ..- attr(*, "units")= chr "secs"# Examine head of time column
head(akl_hourly$time)## 00:00:00
## 00:30:00
## 01:00:00
## 01:30:00
## 02:00:00
## 02:30:00# A plot using just time
ggplot(akl_hourly, aes(x = time, y = temperature)) +
geom_line(aes(group = make_date(year, month, mday)), alpha = 0.2)
library(microbenchmark)
library(fasttime)
# Examine structure of dates
dates <- paste0(gsub(" ", "T", as.character(akl_hourly$date_utc)), "Z")
str(dates)## chr [1:17454] "2015-12-31T11:00:00Z" "2015-12-31T11:30:00Z" ...# Use fastPOSIXct() to parse dates
fastPOSIXct(dates) %>% str()## POSIXct[1:17454], format: "2015-12-31 05:00:00" "2015-12-31 05:30:00" ...# Compare speed of fastPOSIXct() to ymd_hms()
microbenchmark(
ymd_hms = ymd_hms(dates),
fasttime = fastPOSIXct(dates),
times = 20)## Unit: milliseconds
## expr min lq mean median uq max neval
## ymd_hms 20.387000 23.975696 27.81115 26.755940 30.550697 39.83252 20
## fasttime 1.699805 1.782901 2.66241 1.910208 2.699714 10.55411 20
## cld
## b
## a# Head of dates
head(dates)## [1] "2015-12-31T11:00:00Z" "2015-12-31T11:30:00Z" "2015-12-31T12:00:00Z"
## [4] "2015-12-31T12:30:00Z" "2015-12-31T13:00:00Z" "2015-12-31T13:30:00Z"# Parse dates with fast_strptime
fast_strptime(dates,
format = "%Y-%m-%dT%H:%M:%SZ") %>% str()## POSIXlt[1:17454], format: "2015-12-31 11:00:00" "2015-12-31 11:30:00" ...# Comparse speed to ymd_hms() and fasttime
microbenchmark(
ymd_hms = ymd_hms(dates),
fasttime = fastPOSIXct(dates),
fast_strptime = fast_strptime(dates,
format = "%Y-%m-%dT%H:%M:%SZ"),
times = 20)## Unit: milliseconds
## expr min lq mean median uq
## ymd_hms 20.495557 24.935932 40.443395 26.790282 34.311110
## fasttime 1.639803 1.703161 2.038030 1.790993 1.973172
## fast_strptime 1.361897 1.429203 2.072649 1.587893 1.868167
## max neval cld
## 232.550750 20 b
## 3.676332 20 a
## 8.872856 20 afinished <- "I finished 'Dates and Times in R' on Thursday, September 20, 2017!"
# Create a stamp based on "Sep 20 2017"
date_stamp <- stamp("September 20, 2017", orders="mdy")## Multiple formats matched: "%Om %d, %Y"(1), "%B %d, %Y"(1)## Using: "%B %d, %Y"# Print date_stamp
date_stamp## function (x, locale = "English_United States.1252")
## {
## {
## old_lc_time <- Sys.getlocale("LC_TIME")
## if (old_lc_time != locale) {
## Sys.setlocale("LC_TIME", locale)
## on.exit(Sys.setlocale("LC_TIME", old_lc_time))
## }
## }
## format(x, format = "%B %d, %Y")
## }
## <environment: 0x00000000119df538># Call date_stamp on today()
date_stamp(today())## [1] "April 06, 2018"# Create and call a stamp based on "09/20/2017"
stamp("09/20/2017", orders="mdy")(today())## Multiple formats matched: "%Om/%d/%Y"(1), "%m/%d/%Y"(1)## Using: "%Om/%d/%Y"## [1] "04/06/2018"# Use string finished for stamp()
stamp(finished, orders="amdy")(today())## Multiple formats matched: "I finished 'Dates and Times in R' on %A, %B %d, %Y!"(1), "I finished 'Dates and Times in R' on %A, %Om %d, %Y!"(0)## Using: "I finished 'Dates and Times in R' on %A, %B %d, %Y!"## [1] "I finished 'Dates and Times in R' on Friday, April 06, 2018!"Scalable Data Processing in R
Chapter 1 - Working with Increasingly Large Data Sets
What is scalable data processing?:
- Working with data that is too large for one computer
- Scalable code lets you work in parallel, and use resources as they become available
- Data sets are frequently much bigger than available RAM, which is a challenge since R by default runs using R
- “R is not well suited to working with data larger than 10%-20% of a computer’s RAM” - The R Installation and Administration Manual
- When a computer runs out of RAM, it “swaps” to the hard drive, vastly slowing down the calculations
- “R is not well suited to working with data larger than 10%-20% of a computer’s RAM” - The R Installation and Administration Manual
- A more scalable solution is as follows
- Move a subset of data in to RAM
- Process the subset
- Keep the results and discard the subset
- Move a subset of data in to RAM
- Code may be slow due to complexity of calculations
- Consider the disk operations needed
- Consider the disk operations needed
- Benchmarking using microbenchmark() can be critical
Working with “out of core” objects using the Bigmemory Project:
- Package “bigmemory” was written by Kane (instructor for this course) to store, manipulate, and process matrices exceeding RAM
- Core object is a big.matrix and it is designed to manage situations where disk space is much greater than RAM
- The process of moving data to RAM only when needed is called “out of core” processing
- Core object is a big.matrix and it is designed to manage situations where disk space is much greater than RAM
- By default, a big.matrix keeps data on the disk, only moving the data to RAM as needed
- The movements to/from RAM are implicit, which is to say that they are managed by the package
- Only a single import is needed
- The movements to/from RAM are implicit, which is to say that they are managed by the package
- The big.matrix is created using big.matrix(nrow=, ncol=, init=, type=, backingfile=, descriptorfile=)
- The nrow, ncol are the same as matrix(), while init is the initial value to assign everywhere and type is a quoted type such as “double” or “integer”
- The backingfile is a quoted file name that will hold the binary representation of the big.matrix on the disk, with extension .bin
- The descriptorfile is a quoted file name that will hold some metadata such as the number of rows/columns, name, and the like
- The nrow, ncol are the same as matrix(), while init is the initial value to assign everywhere and type is a quoted type such as “double” or “integer”
- Supposing that x is a big,matrix, then the default print(x) obtained by x on the command line is to show a few slots/pointers
- To have contents of x printed, use x[ , ]
- Assignments can be made using x[myRow, myColumn] <- myValue
- To have contents of x printed, use x[ , ]
- The read.big.matrix() function is meant to look similar to read.table() but, in addition, needs to know:
- what type of numeric values you want to read (“char”, “short”, “integer”, “double”)
- name of the file that will hold the matrix’s data (the backing file)
- name of the file to hold information about the matrix (a descriptor file)
- Result will be a file on the disk holding the value read in along with a descriptor file which holds extra information (like the number of columns and rows) about the resulting big.matrix object
- what type of numeric values you want to read (“char”, “short”, “integer”, “double”)
- A final advantage to using big.matrix is that if you know how to use R’s matrices, then you know how to use a big.matrix
- You can subset columns and rows just as you would a regular matrix, using a numeric or character vector and the object returned is an R matrix
- Likewise, assignments are the same as with R matrices and after those assignments are made they are stored on disk and can be used in the current and future R sessions
- One thing to remember is that $ is not valid for getting a column of either a matrix or a big.matrix
- You can subset columns and rows just as you would a regular matrix, using a numeric or character vector and the object returned is an R matrix
References vs. Copies:
- Can subset and make assignments to a big.matrix much like a matrix
- There are a few key differences between a big.matrix and a matrix
- big.matrix is stored on the disk (persists across R sessions, can be shared across R sessions)
- R typically makes copies during assignment, which is why changing a variable inside a function (playing with the copy) has no impact on the variable outside the function
- However, some objects such as environments are not copied, so modifying them inside a function modified them globally (outside the function) also
- The big.matrix is not copied, and is instead a reference object; thus, you have to explicitly request a copy, which means 1) you have more control, but 2) you need to be more careful
- big.matrix is stored on the disk (persists across R sessions, can be shared across R sessions)
- The reference vs. copy for big.matrix objects seems in some ways similar to Python
- a <- b will set a to reference the same data as b; changing a or changing b means changing both
- a <- deepcopy() will produce a copy of a and assign it to b; much like a = b[:] in Python
- a <- b will set a to reference the same data as b; changing a or changing b means changing both
Example code includes:
# Load the microbenchmark package
library(microbenchmark)
# Compare the timings for sorting different sizes of vector
mb <- microbenchmark(
# Sort a random normal vector length 1e5
"1e5" = sort(rnorm(1e5)),
# Sort a random normal vector length 2.5e5
"2.5e5" = sort(rnorm(2.5e5)),
# Sort a random normal vector length 5e5
"5e5" = sort(rnorm(5e5)),
"7.5e5" = sort(rnorm(7.5e5)),
"1e6" = sort(rnorm(1e6)),
times = 10
)
# Plot the resulting benchmark object
plot(mb)
# Load the bigmemory package
library(bigmemory)
# Create the big.matrix object: x
x <- read.big.matrix("./RInputFiles/mortgage-sample.csv", header = TRUE,
type = "integer",
backingfile = "mortgage-sample.bin",
descriptorfile = "mortgage-sample.desc")
# Find the dimensions of x
dim(x)## [1] 70000 16# Attach mortgage-sample.desc
mort <- attach.big.matrix("mortgage-sample.desc")
# Find the dimensions of mort
dim(mort)## [1] 70000 16# Look at the first 6 rows of mort
head(mort)## enterprise record_number msa perc_minority tract_income_ratio
## [1,] 1 566 1 1 3
## [2,] 1 116 1 3 2
## [3,] 1 239 1 2 2
## [4,] 1 62 1 2 3
## [5,] 1 106 1 2 3
## [6,] 1 759 1 3 3
## borrower_income_ratio loan_purpose federal_guarantee borrower_race
## [1,] 1 2 4 3
## [2,] 1 2 4 5
## [3,] 3 8 4 5
## [4,] 3 2 4 5
## [5,] 3 2 4 9
## [6,] 2 2 4 9
## co_borrower_race borrower_gender co_borrower_gender num_units
## [1,] 9 2 4 1
## [2,] 9 1 4 1
## [3,] 5 1 2 1
## [4,] 9 2 4 1
## [5,] 9 3 4 1
## [6,] 9 1 2 2
## affordability year type
## [1,] 3 2010 1
## [2,] 3 2008 1
## [3,] 4 2014 0
## [4,] 4 2009 1
## [5,] 4 2013 1
## [6,] 4 2010 1# Create mort
mort <- attach.big.matrix("mortgage-sample.desc")
# Look at the first 3 rows
mort[1:3, ]## enterprise record_number msa perc_minority tract_income_ratio
## [1,] 1 566 1 1 3
## [2,] 1 116 1 3 2
## [3,] 1 239 1 2 2
## borrower_income_ratio loan_purpose federal_guarantee borrower_race
## [1,] 1 2 4 3
## [2,] 1 2 4 5
## [3,] 3 8 4 5
## co_borrower_race borrower_gender co_borrower_gender num_units
## [1,] 9 2 4 1
## [2,] 9 1 4 1
## [3,] 5 1 2 1
## affordability year type
## [1,] 3 2010 1
## [2,] 3 2008 1
## [3,] 4 2014 0# Create a table of the number of mortgages for each year in the data set
table(mort[, "year"])##
## 2008 2009 2010 2011 2012 2013 2014 2015
## 8468 11101 8836 7996 10935 10216 5714 6734a <- getLoadedDLLs()
length(a)## [1] 39R.utils::gcDLLs()## named list()a <- getLoadedDLLs()
length(a)## [1] 39# Load the biganalytics package (error in loading to Knit file, works OK otherwise)
library(biganalytics)## Loading required package: foreach## Loading required package: biglm## Loading required package: DBI# Get the column means of mort
colmean(mort)## enterprise record_number msa
## 1.3814571 499.9080571 0.8943571
## perc_minority tract_income_ratio borrower_income_ratio
## 1.9701857 2.3431571 2.6898857
## loan_purpose federal_guarantee borrower_race
## 3.7670143 3.9840857 5.3572429
## co_borrower_race borrower_gender co_borrower_gender
## 7.0002714 1.4590714 3.0494857
## num_units affordability year
## 1.0398143 4.2863429 2011.2714714
## type
## 0.5300429# Use biganalytics' summary function to get a summary of the data
summary(mort)## min max mean NAs
## enterprise 1.0000000 2.0000000 1.3814571 0.0000000
## record_number 0.0000000 999.0000000 499.9080571 0.0000000
## msa 0.0000000 1.0000000 0.8943571 0.0000000
## perc_minority 1.0000000 9.0000000 1.9701857 0.0000000
## tract_income_ratio 1.0000000 9.0000000 2.3431571 0.0000000
## borrower_income_ratio 1.0000000 9.0000000 2.6898857 0.0000000
## loan_purpose 1.0000000 9.0000000 3.7670143 0.0000000
## federal_guarantee 1.0000000 4.0000000 3.9840857 0.0000000
## borrower_race 1.0000000 9.0000000 5.3572429 0.0000000
## co_borrower_race 1.0000000 9.0000000 7.0002714 0.0000000
## borrower_gender 1.0000000 9.0000000 1.4590714 0.0000000
## co_borrower_gender 1.0000000 9.0000000 3.0494857 0.0000000
## num_units 1.0000000 4.0000000 1.0398143 0.0000000
## affordability 0.0000000 9.0000000 4.2863429 0.0000000
## year 2008.0000000 2015.0000000 2011.2714714 0.0000000
## type 0.0000000 1.0000000 0.5300429 0.0000000# Use deepcopy() to create first_three
first_three <- deepcopy(mort, cols = 1:3,
backingfile = "first_three.bin",
descriptorfile = "first_three.desc")
# Set first_three_2 equal to first_three
first_three_2 <- first_three
# Set the value in the first row and first column of first_three to NA
first_three[1, 1] <- NA
# Verify the change shows up in first_three_2
first_three_2[1, 1]## [1] NA# but not in mort
mort[1, 1]## [1] 1Chapter 2 - Processing and Analyzing Data with bigmemory
The Bigmemory Suite of Packages:
- Many packages have been designed to work together with a big.matrix object
- biganalytics - summarizing
- bigtabulate - split and tabulate (includes the bigtable(x, quotedColumnVector))
- bigalgebra - linear algenra
- bigpca - PCA
- bigFastLM - linear regressions
- biglasso - lasso regressions
- bigrf - random forests
- biganalytics - summarizing
- FHFA Dataset has data about millions of mortgages - difference in ownership rates, defaults, etc.
- Course will use a 70,000 record subset
- Raw data (full 2.5 GB dataset) available at FHFA (fhfa.gov)
- Code works the same on subsets and full data sets
- Course will use a 70,000 record subset
Split-Apply-Combine (aka Split-Compute-Combine), run in this course using split() Map() Reduce():
- The split() function partitions the data, whether randomly or based on a factor variable
- split(myData, myFactor) will produce a list, with each element of the list containing the requested data (one per myFactor)
- split(myData, myFactor) will produce a list, with each element of the list containing the requested data (one per myFactor)
- The Map() function processes each of the partitions
- Map(myFunction, mySplitList) will apply the myFunction to each of the items in the mySplitList, with the output a list named like mySplitList
- Map(myFunction, mySplitList) will apply the myFunction to each of the items in the mySplitList, with the output a list named like mySplitList
- The Reduce() function combines the (typically processed) data from a list
- Reduce(myFunction, myMapList) will apply the myFunction while combining the items in myMapList
- A common function might be rbind or ‘+’ (add them up)
- Reduce(myFunction, myMapList) will apply the myFunction while combining the items in myMapList
Visualize results using tidyverse:
- The pipe (%>%) operator works well with many of the big.matrix functions, since the first argument is a dataset
- Can combine some of big.matrix processing outputs with standard packages like dplyr and tidyr and ggplot
Limitations of bigmemory - process is useful for dense, numeric matrices that can be stored on hard disk:
- Underlying structures are compatible with low-level linear algebra libraries for fast fitting
- If you have different column types, you can try the ff package (similar to bigmemory but includes structures like a data.frame)
- The bigmemory object is said to be “random access”, which means it is equally easy to get access to any specific component
- There are some big drawbacks to the “random access” capabilities, however
- Cannot add rows or columns - need to create an entirely new object and port over the relevant data
- Need enough disk space to hold the entire matrix in a block
- Can instead use other tools to process data using a “continuous chunks” approach - discussed in the next chapter
- Cannot add rows or columns - need to create an entirely new object and port over the relevant data
Example code includes:
library(bigtabulate)
library(tidyr)
library(ggplot2)
library(biganalytics)
library(dplyr)
race_cat <- c('Native Am', 'Asian', 'Black', 'Pacific Is', 'White', 'Two or More', 'Hispanic', 'Not Avail')
# Call bigtable to create a variable called race_table
race_table <- bigtable(mort, "borrower_race")
# Rename the elements of race_table
names(race_table) <- race_cat
race_table## Native Am Asian Black Pacific Is White Two or More
## 143 4438 2020 195 50006 528
## Hispanic Not Avail
## 4040 8630# Create a table of the borrower race by year
race_year_table <- bigtable(mort, c("borrower_race", "year"))
# Convert rydf to a data frame
rydf <- as.data.frame(race_year_table)
# Create the new column Race
rydf$Race <- race_cat
# Let's see what it looks like
rydf## 2008 2009 2010 2011 2012 2013 2014 2015 Race
## 1 11 18 13 16 15 12 29 29 Native Am
## 2 384 583 603 568 770 673 369 488 Asian
## 3 363 320 209 204 258 312 185 169 Black
## 4 33 38 21 13 28 22 17 23 Pacific Is
## 5 5552 7739 6301 5746 8192 7535 4110 4831 White
## 6 43 85 65 58 89 78 46 64 Two or More
## 7 577 563 384 378 574 613 439 512 Hispanic
## 9 1505 1755 1240 1013 1009 971 519 618 Not Availfemale_residence_prop <- function(x, rows) {
x_subset <- x[rows, ]
# Find the proporation of female borrowers in urban areas
prop_female_urban <- sum(x_subset[, "borrower_gender"] == 2 &
x_subset[, "msa"] == 1) /
sum(x_subset[, "msa"] == 1)
# Find the proporation of female borrowers in rural areas
prop_female_rural <- sum(x_subset[, "borrower_gender"] == 2 &
x_subset[, "msa"] == 0) /
sum(x_subset[, "msa"] == 0)
c(prop_female_urban, prop_female_rural)
}
# Find the proportion of female borrowers in 2015
female_residence_prop(mort, mort[, "year"] == 2015)## [1] 0.2737439 0.2304965# Split the row numbers of the mortage data by year
spl <- split(1:nrow(mort), mort[, "year"])
# Call str on spl
str(spl)## List of 8
## $ 2008: int [1:8468] 2 8 15 17 18 28 35 40 42 47 ...
## $ 2009: int [1:11101] 4 13 25 31 43 49 52 56 67 68 ...
## $ 2010: int [1:8836] 1 6 7 10 21 23 24 27 29 38 ...
## $ 2011: int [1:7996] 11 20 37 46 53 57 73 83 86 87 ...
## $ 2012: int [1:10935] 14 16 26 30 32 33 48 69 81 94 ...
## $ 2013: int [1:10216] 5 9 19 22 36 44 55 58 72 74 ...
## $ 2014: int [1:5714] 3 12 50 60 64 66 103 114 122 130 ...
## $ 2015: int [1:6734] 34 41 54 61 62 65 82 91 102 135 ...# For each of the row splits, find the female residence proportion
all_years <- Map(function(rows) female_residence_prop(mort, rows), spl)
# Call str on all_years
str(all_years)## List of 8
## $ 2008: num [1:2] 0.275 0.204
## $ 2009: num [1:2] 0.244 0.2
## $ 2010: num [1:2] 0.241 0.201
## $ 2011: num [1:2] 0.252 0.241
## $ 2012: num [1:2] 0.244 0.21
## $ 2013: num [1:2] 0.275 0.257
## $ 2014: num [1:2] 0.289 0.268
## $ 2015: num [1:2] 0.274 0.23# Collect the results as rows in a matrix
prop_female <- Reduce(rbind, all_years)
# Rename the row and column names
dimnames(prop_female) <- list(names(all_years), c("prop_female_urban", "prop_femal_rural"))
# View the matrix
prop_female## prop_female_urban prop_femal_rural
## 2008 0.2748514 0.2039474
## 2009 0.2441074 0.2002978
## 2010 0.2413881 0.2014028
## 2011 0.2520644 0.2408931
## 2012 0.2438950 0.2101313
## 2013 0.2751059 0.2567164
## 2014 0.2886756 0.2678571
## 2015 0.2737439 0.2304965# Convert prop_female to a data frame
prop_female_df <- as.data.frame(prop_female)
# Add a new column Year
prop_female_df$Year <- row.names(prop_female_df)
# Call gather on prop_female_df
prop_female_long <- gather(prop_female_df, Region, Prop, -Year)
# Create a line plot
ggplot(prop_female_long, aes(x = Year, y = Prop, group = Region, color = Region)) +
geom_line()
# Call summary on mort
summary(mort)## min max mean NAs
## enterprise 1.0000000 2.0000000 1.3814571 0.0000000
## record_number 0.0000000 999.0000000 499.9080571 0.0000000
## msa 0.0000000 1.0000000 0.8943571 0.0000000
## perc_minority 1.0000000 9.0000000 1.9701857 0.0000000
## tract_income_ratio 1.0000000 9.0000000 2.3431571 0.0000000
## borrower_income_ratio 1.0000000 9.0000000 2.6898857 0.0000000
## loan_purpose 1.0000000 9.0000000 3.7670143 0.0000000
## federal_guarantee 1.0000000 4.0000000 3.9840857 0.0000000
## borrower_race 1.0000000 9.0000000 5.3572429 0.0000000
## co_borrower_race 1.0000000 9.0000000 7.0002714 0.0000000
## borrower_gender 1.0000000 9.0000000 1.4590714 0.0000000
## co_borrower_gender 1.0000000 9.0000000 3.0494857 0.0000000
## num_units 1.0000000 4.0000000 1.0398143 0.0000000
## affordability 0.0000000 9.0000000 4.2863429 0.0000000
## year 2008.0000000 2015.0000000 2011.2714714 0.0000000
## type 0.0000000 1.0000000 0.5300429 0.0000000bir_df_wide <- bigtable(mort, c("borrower_income_ratio", "year")) %>%
as.data.frame() %>%
tibble::rownames_to_column() %>%
filter(rowname %in% c(1, 2, 3)) %>%
select(-rowname) %>%
# Create a new column called BIR with the corresponding table categories
mutate(BIR = c(">=0,<=50%", ">50, <=80%", ">80%"))
bir_df_wide## 2008 2009 2010 2011 2012 2013 2014 2015 BIR
## 1 1205 1473 600 620 745 725 401 380 >=0,<=50%
## 2 2095 2791 1554 1421 1819 1861 1032 1145 >50, <=80%
## 3 4844 6707 6609 5934 8338 7559 4255 5169 >80%bir_df_wide %>%
# Transform the wide-formatted data.frame into the long format
gather(Year, Count, -BIR) %>%
# Use ggplot to create a line plot
ggplot(aes(x = Year, y = Count, group = BIR, color = BIR)) +
geom_line()
Chapter 3 - Working with iotools
Introduction to chunk-wise processing - solution to challenges from bigmemory:
- The iotools allows for processing the data in “chunks”, allowing for data frames, data across many machines, and the like
- Can process chunks either sequentially (keep as needed after each chunk runs) or independently
- Independent processing is typically harder to code (final result must be combined), but allows for parallel processing
- Independent processing is typically harder to code (final result must be combined), but allows for parallel processing
- Sometimes Split-Apply-Combine cannot be made to work, such as trying to find a median (even keeping some extra data per chunk – such as sum and count when end goal is mean – will not work)
- Fortunately, most regressions can be successfully run using the Split-Apply-Combine methodology
- Fortunately, most regressions can be successfully run using the Split-Apply-Combine methodology
- An operation that gives the same answer whether you apply it to an entire data set or to chunks of a data set and then on the results on the chunks is sometimes called foldable
- The max() and min() operations are an example of this
First look at iotools: Importing data:
- Basic components of chunk-wise processing include 1) load pieces of data, 2) convert to native objects, 3) perform computation and store results, and 4) repeated as needed until finished
- Loading data often takes more time than processing the data (retrieval from disk and conversion to readable formats)
- The iotools package is designed to separate the physical loading of data and the parsing of data in to R objects for better flexibility and performance
- readAsRaw() reads the entire data in to a raw vector
- read.chunk() reads the data in chunks in to a raw vector
- readAsRaw() reads the entire data in to a raw vector
- The iotools can then parse the data in to either a matrix or a data frame
- mstrsplit() converts raw data in to a matrix
- dstrsplit() converts raw data in to a data frame
- read.delim.raw() = readAsRaw() + dstrsplit()
- mstrsplit() converts raw data in to a matrix
- Processing contiguous chunks means there is no need to have read all the data in advance (such as to create the spl vector by 1:nrows by myVar)
- When processing a sequence of contiguous chunks of data on a hard drive, iotools can turn a raw object into a data.frame or matrix while - at the same time - retrieving the next chunk of data
- These optimizations allow iotools to quickly process very large files
Using chunk.apply - effectively moves away from what is functionally a “for loop” to allow better parallel processing:
- iotools is the basis of hmr which allows for running R on TB of data using Hadoop
- The general usage is chunk.apply(myFile=, myFunction=, CH.MAX.SIZE=) # this will apply myFunction across chunks of size CH.MAX.SIZE in myFile
- Output will be a matrix where each row is one of the chunks and each column is one of (or the only) output from myFunction for that chunk
- There is an optional parallel= option; the argument supplied is the number of parallel clusters to be used
- Output will be a matrix where each row is one of the chunks and each column is one of (or the only) output from myFunction for that chunk
- By default, chunk.apply() aggregates the processed data using the rbind() function
- This means that you can create a table from each of the chunks and then add up the rows of the resulting matrix to get the total counts for the table
- This means that you can create a table from each of the chunks and then add up the rows of the resulting matrix to get the total counts for the table
- When the parallel parameter is set to a value greater than one on Linux and Unix machine (including the Mac) multiple processes read and process data at the same time thereby reducing the execution time
- On Windows the parallel parameter is ignored
Example code includes:
foldable_range <- function(x) {
if (is.list(x)) {
# If x is a list then reduce it by the min and max of each element in the list
c(Reduce(min, x), Reduce(max, x))
} else {
# Otherwise, assume it's a vector and find it's range
range(x)
}
}
# Verify that foldable_range() works on the record_number column
foldable_range(mort[, "record_number"])## [1] 0 999# Split the mortgage data by year
spl <- split(1:nrow(mort), mort[, "year"])
# Use foldable_range() to get the range of the record numbers
foldable_range(Map(function(s) foldable_range(mort[s, "record_number"]), spl))## [1] 0 999# Load the iotools and microbenchmark packages
library(iotools)
library(microbenchmark)
# Time the reading of files
microbenchmark(
# Time the reading of a file using read.delim five times
read.delim("./RInputFiles/mortgage-sample.csv", header = FALSE, sep = ","),
# Time the reading of a file using read.delim.raw five times
read.delim.raw("./RInputFiles/mortgage-sample.csv", header = FALSE, sep = ","),
times = 5
)## Unit: milliseconds
## expr
## read.delim("./RInputFiles/mortgage-sample.csv", header = FALSE, sep = ",")
## read.delim.raw("./RInputFiles/mortgage-sample.csv", header = FALSE, sep = ",")
## min lq mean median uq max neval cld
## 483.29446 505.05836 599.8070 586.8768 659.8168 763.9885 5 b
## 86.29924 88.14353 181.2442 100.0256 133.5649 498.1877 5 a# Read mortgage-sample.csv as a raw vector
raw_file_content <- readAsRaw("./RInputFiles/mortgage-sample.csv")
# Convert the raw vector contents to a matrix
mort_mat <- mstrsplit(raw_file_content, sep = ",", type = "integer", skip = 1)
# Look at the first 6 rows
head(mort_mat)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 1 566 1 1 3 1 2 4 3 9 2 4 1
## [2,] 1 116 1 3 2 1 2 4 5 9 1 4 1
## [3,] 1 239 1 2 2 3 8 4 5 5 1 2 1
## [4,] 1 62 1 2 3 3 2 4 5 9 2 4 1
## [5,] 1 106 1 2 3 3 2 4 9 9 3 4 1
## [6,] 1 759 1 3 3 2 2 4 9 9 1 2 2
## [,14] [,15] [,16]
## [1,] 3 2010 1
## [2,] 3 2008 1
## [3,] 4 2014 0
## [4,] 4 2009 1
## [5,] 4 2013 1
## [6,] 4 2010 1# Convert the raw file contents to a data.frame
mort_df <- dstrsplit(raw_file_content, sep = ",", col_types = rep("integer", 16), skip = 1)
# Look at the first 6 rows
head(mort_df)## V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16
## 1 1 566 1 1 3 1 2 4 3 9 2 4 1 3 2010 1
## 2 1 116 1 3 2 1 2 4 5 9 1 4 1 3 2008 1
## 3 1 239 1 2 2 3 8 4 5 5 1 2 1 4 2014 0
## 4 1 62 1 2 3 3 2 4 5 9 2 4 1 4 2009 1
## 5 1 106 1 2 3 3 2 4 9 9 3 4 1 4 2013 1
## 6 1 759 1 3 3 2 2 4 9 9 1 2 2 4 2010 1# We have created a file connection fc to the "mortgage-sample.csv" file and read in the first line to get rid of the header.
# Define the function to apply to each chunk
make_table <- function(chunk) {
# Read each chunk as a matrix
x <- mstrsplit(chunk, type = "integer", sep = ",")
# Create a table of the number of borrowers (column 3) for each chunk
table(x[, 3])
}
# Create a file connection to mortgage-sample.csv
fc <- file("./RInputFiles/mortgage-sample.csv", "rb")
# Read the first line to get rid of the header
(col_names <- readLines(fc, n = 1))## [1] "\"enterprise\",\"record_number\",\"msa\",\"perc_minority\",\"tract_income_ratio\",\"borrower_income_ratio\",\"loan_purpose\",\"federal_guarantee\",\"borrower_race\",\"co_borrower_race\",\"borrower_gender\",\"co_borrower_gender\",\"num_units\",\"affordability\",\"year\",\"type\""(col_names <- lapply(str_split(col_names, '\\",\\"'), FUN=function(x) { str_replace(x, '\\"', '') })[[1]])## [1] "enterprise" "record_number"
## [3] "msa" "perc_minority"
## [5] "tract_income_ratio" "borrower_income_ratio"
## [7] "loan_purpose" "federal_guarantee"
## [9] "borrower_race" "co_borrower_race"
## [11] "borrower_gender" "co_borrower_gender"
## [13] "num_units" "affordability"
## [15] "year" "type"# Read the data in chunks
counts <- chunk.apply(fc, make_table, CH.MAX.SIZE = 1e5)
# Close the file connection
close(fc)
# Print counts
counts## 0 1
## [1,] 309 2401
## [2,] 289 2422
## [3,] 266 2444
## [4,] 300 2410
## [5,] 279 2431
## [6,] 310 2400
## [7,] 274 2436
## [8,] 283 2428
## [9,] 259 2452
## [10,] 287 2423
## [11,] 288 2423
## [12,] 283 2428
## [13,] 271 2439
## [14,] 299 2411
## [15,] 294 2416
## [16,] 305 2405
## [17,] 280 2431
## [18,] 275 2435
## [19,] 303 2407
## [20,] 279 2431
## [21,] 296 2414
## [22,] 294 2417
## [23,] 288 2424
## [24,] 264 2446
## [25,] 292 2418
## [26,] 228 2013# Sum up the chunks
colSums(counts)## 0 1
## 7395 62605msa_map <- c("rural", "urban")
# Define the function to apply to each chunk
make_msa_table <- function(chunk) {
# Read each chunk as a data frame
x <- dstrsplit(chunk, col_types = rep("integer", length(col_names)), sep = ",")
# Set the column names of the data frame that's been read
colnames(x) <- col_names
# Create new column, msa_pretty, with a string description of where the borrower lives
x$msa_pretty <- msa_map[x$msa + 1]
# Create a table from the msa_pretty column
table(x$msa_pretty)
}
# Create a file connection to mortgage-sample.csv
fc <- file("./RInputFiles/mortgage-sample.csv", "rb")
# Read the first line to get rid of the header
readLines(fc, n = 1)## [1] "\"enterprise\",\"record_number\",\"msa\",\"perc_minority\",\"tract_income_ratio\",\"borrower_income_ratio\",\"loan_purpose\",\"federal_guarantee\",\"borrower_race\",\"co_borrower_race\",\"borrower_gender\",\"co_borrower_gender\",\"num_units\",\"affordability\",\"year\",\"type\""# Read the data in chunks
counts <- chunk.apply(fc, make_msa_table, CH.MAX.SIZE = 1e5)
# Close the file connection
close(fc)
# Aggregate the counts as before
colSums(counts)## rural urban
## 7395 62605iotools_read_fun <- function(parallel) {
fc <- file("./RInputFiles/mortgage-sample.csv", "rb")
readLines(fc, n = 1)
chunk.apply(fc, make_msa_table,
CH.MAX.SIZE = 1e5, parallel = parallel)
close(fc)
}
# Benchmark the new function
microbenchmark(
# Use one process
iotools_read_fun(1),
# Use three processes
iotools_read_fun(3),
times = 20
)## Unit: milliseconds
## expr min lq mean median uq max
## iotools_read_fun(1) 102.8291 116.6638 147.6507 124.0177 193.0392 231.5074
## iotools_read_fun(3) 111.2729 126.3639 193.1596 186.3738 223.7989 378.9957
## neval cld
## 20 a
## 20 bChapter 4 - Case Study: Preliminary Analysis of Housing Data
Overview of types of analysis for this chapter:
- Compare proportions of people receiving mortgages
- Amount of “missingness” in the data
- Changes in 1) mortgage demographic proportions over time, and 2) city vs. rural mortgages, and 3) proportions of federally insured loans
Are the data missing at random?
- Missing data is pervasive, including in this housing dataset
- Three components of missing data
- Missing Completely at Random (MCAR) - no way to predict where/what, meaning rows with missing data can just be dropped
- Missing at Random (MAR) - missingness is dependent on variables in the dataset, meaning that multiple imputation can be successful
- Missing Not At Random (MNAR) - typically due to deterministic relationships between missing data and other variables, beyond the scope of this course
- Missing Completely at Random (MCAR) - no way to predict where/what, meaning rows with missing data can just be dropped
- Assumption for this exercise will be that data are checked for MAR and assumed to be MCAR if they are not MAR
- For each column, recode the column as a 1/0 for missing, then run a logit on all the other variables
- If the other variables have a statistically significant prediction effect on the 1/0 column, then that column is MAR rather than MCAR
- Need to have a smart p-value for significance depending on number of regressions that have been run
- For each column, recode the column as a 1/0 for missing, then run a logit on all the other variables
Analyzing the Housing Data:
- Adjusted counts - adjusting group sizes allows you to compare different groups as though they were the same size
- Proportional change can show growth (or decline) of groups over time
Borrower Lending Trends: City vs. Rural:
- Looking at city (MSA == 1) vs rural
- Looking at federally guaranteed loans
- Can use Borrower Income Ratio (borrower income divided by median income in the area)
Wrap up:
- Split-Compute-Combine (aka Split-Apply-Combine) as enabled by bigmemory and iotools
- Operations can be run on a single machine in series, a single machine in parallel, or across multiple machines
- Summary of the bigmemory approach
- Good for dense, large matrices that might otherwise overhwlem RAM
- Looks like a regular R matrix
- Good for dense, large matrices that might otherwise overhwlem RAM
- Summary of the iotools approach:
- Good for much larger data that can be processed in sequential chunks
- More flexible than bigmemory in that it can handle data frames and files saved on multiple disks
- Good for much larger data that can be processed in sequential chunks
Example code includes:
# Create a table of borrower_race column
race_table <- bigtable(mort, "borrower_race")
# Rename the elements
names(race_table) <- race_cat[as.numeric(names(race_table))]
# Find the proportion
race_table[1:7] / sum(race_table[1:7])## Native Am Asian Black Pacific Is White Two or More
## 0.002330129 0.072315464 0.032915105 0.003177448 0.814828092 0.008603552
## Hispanic
## 0.065830210mort_names <- col_names
# Create table of the borrower_race
race_table_chunks <- chunk.apply(
"./RInputFiles/mortgage-sample.csv", function(chunk) {
x <- mstrsplit(chunk, sep = ",", type = "integer")
colnames(x) <- mort_names
table(x[, "borrower_race"])
}, CH.MAX.SIZE = 1e5)
# Add up the columns
race_table <- colSums(race_table_chunks)
# Find the proportion
borrower_proportion <- race_table[1:7] / sum(race_table[1:7])
pop_proportion <- c(0.009, 0.048, 0.126, 0.002, 0.724, 0.029, 0.163)
names(pop_proportion) <- race_cat[1:7]
# Create the matrix
matrix(c(pop_proportion, borrower_proportion), byrow = TRUE, nrow = 2,
dimnames = list(c("Population Proportion", "Borrower Proportion"), race_cat[1:7]))## Native Am Asian Black Pacific Is
## Population Proportion 0.009000000 0.04800000 0.12600000 0.002000000
## Borrower Proportion 0.002330129 0.07231546 0.03291511 0.003177448
## White Two or More Hispanic
## Population Proportion 0.7240000 0.029000000 0.16300000
## Borrower Proportion 0.8148281 0.008603552 0.06583021# Create a variable indicating if borrower_race is missing in the mortgage data
borrower_race_ind <- mort[, "borrower_race"] == 9
# Create a factor variable indicating the affordability
affordability_factor <- factor(mort[, "affordability"])
# Perform a logistic regression
summary(glm(borrower_race_ind ~ affordability_factor, family = binomial))## Warning: closing unused connection 5 (./RInputFiles/mortgage-sample.csv)##
## Call:
## glm(formula = borrower_race_ind ~ affordability_factor, family = binomial)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5969 -0.5016 -0.5016 -0.5016 2.0867
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.7478 0.1376 -12.701 <2e-16 ***
## affordability_factor1 -0.2241 0.1536 -1.459 0.1447
## affordability_factor2 -0.3090 0.1609 -1.920 0.0548 .
## affordability_factor3 -0.2094 0.1446 -1.448 0.1476
## affordability_factor4 -0.2619 0.1383 -1.894 0.0582 .
## affordability_factor9 0.1131 0.1413 0.800 0.4235
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 52279 on 69999 degrees of freedom
## Residual deviance: 52166 on 69994 degrees of freedom
## AIC: 52178
##
## Number of Fisher Scoring iterations: 4# Open a connection to the file and skip the header
fc <- file("./RInputFiles/mortgage-sample.csv", "rb")
readLines(fc, n = 1)## [1] "\"enterprise\",\"record_number\",\"msa\",\"perc_minority\",\"tract_income_ratio\",\"borrower_income_ratio\",\"loan_purpose\",\"federal_guarantee\",\"borrower_race\",\"co_borrower_race\",\"borrower_gender\",\"co_borrower_gender\",\"num_units\",\"affordability\",\"year\",\"type\""# Create a function to read chunks
make_table <- function(chunk) {
# Create a matrix
m <- mstrsplit(chunk, sep = ",", type = "integer")
colnames(m) <- mort_names
# Create the output table
bigtable(m, c("borrower_race", "year"))
}
# Import data using chunk.apply
race_year_table <- chunk.apply(fc, make_table)
# Close connection
close(fc)
# Cast it to a data frame
rydf <- as.data.frame(race_year_table)
# Create a new column Race with race/ethnicity
rydf$Race <- race_cat
# Note: We removed the row corresponding to "Not Avail".
# View rydf
rydf <-
rydf %>%
filter(Race !="Not Avail")
rydf ## 2008 2009 2010 2011 2012 2013 2014 2015 Race
## 1 11 18 13 16 15 12 29 29 Native Am
## 2 384 583 603 568 770 673 369 488 Asian
## 3 363 320 209 204 258 312 185 169 Black
## 4 33 38 21 13 28 22 17 23 Pacific Is
## 5 5552 7739 6301 5746 8192 7535 4110 4831 White
## 6 43 85 65 58 89 78 46 64 Two or More
## 7 577 563 384 378 574 613 439 512 Hispanic# View pop_proportion
pop_proportion## Native Am Asian Black Pacific Is White Two or More
## 0.009 0.048 0.126 0.002 0.724 0.029
## Hispanic
## 0.163# Gather on all variables except Race
rydfl <- gather(rydf, Year, Count, -Race)
# Create a new adjusted count variable
rydfl$Adjusted_Count <- rydfl$Count / pop_proportion[rydfl$Race]
# Plot
ggplot(rydfl, aes(x = Year, y = Adjusted_Count, group = Race, color = Race)) +
geom_line()
# View rydf
rydf## 2008 2009 2010 2011 2012 2013 2014 2015 Race
## 1 11 18 13 16 15 12 29 29 Native Am
## 2 384 583 603 568 770 673 369 488 Asian
## 3 363 320 209 204 258 312 185 169 Black
## 4 33 38 21 13 28 22 17 23 Pacific Is
## 5 5552 7739 6301 5746 8192 7535 4110 4831 White
## 6 43 85 65 58 89 78 46 64 Two or More
## 7 577 563 384 378 574 613 439 512 Hispanic# Normalize the columns
for (i in seq_len(nrow(rydf))) {
rydf[i, 1:8] <- rydf[i, 1:8] / rydf[i, 1]
}
# Convert the data to long format
rydf_long <- gather(rydf, Year, Proportion, -Race)
# Plot
ggplot(rydf_long, aes(x = Year, y = Proportion, group = Race, color = Race)) +
geom_line()
# Open a connection to the file and skip the header
fc <- file("./RInputFiles/mortgage-sample.csv", "rb")
readLines(fc, n = 1)## [1] "\"enterprise\",\"record_number\",\"msa\",\"perc_minority\",\"tract_income_ratio\",\"borrower_income_ratio\",\"loan_purpose\",\"federal_guarantee\",\"borrower_race\",\"co_borrower_race\",\"borrower_gender\",\"co_borrower_gender\",\"num_units\",\"affordability\",\"year\",\"type\""# Create a function to read chunks
make_table <- function(chunk) {
# Create a matrix
m <- mstrsplit(chunk, sep = ",", type = "integer")
colnames(m) <- mort_names
# Create the output table
bigtable(m, c("msa", "year"))
}
# Import data using chunk.apply
msa_year_table <- chunk.apply(fc, make_table)
# Close connection
close(fc)
# Convert to a data frame
df_msa <- as.data.frame(msa_year_table)
# Rename columns
df_msa$MSA <- c("rural", "city")
# Gather on all columns except Year
df_msa_long <- gather(df_msa, Year, Count, -MSA)
# Plot
ggplot(df_msa_long, aes(x = Year, y = Count, group = MSA, color = MSA)) +
geom_line()
# Tabulate borrower_income_ratio and federal_guarantee
ir_by_fg <- bigtable(mort, c("borrower_income_ratio", "federal_guarantee"))
# Label the columns and rows of the table
income_cat <- c('0 <= 50', '50 < 80', '> 80', 'Not Applicable')
guarantee_cat <- c('FHA/VA', 'RHS', 'HECM', 'No Guarantee')
dimnames(ir_by_fg) <- list(income_cat, guarantee_cat)
# For each row in ir_by_fg, divide by the sum of the row
for (i in seq_len(nrow(ir_by_fg))) {
ir_by_fg[i, ] = ir_by_fg[i, ] / sum(ir_by_fg[i, ])
}
# Print
ir_by_fg## FHA/VA RHS HECM No Guarantee
## 0 <= 50 0.008944544 0.0014636526 0.0443974630 0.9451943
## 50 < 80 0.005977548 0.0024055985 0.0026971862 0.9889197
## > 80 0.001113022 0.0002428412 0.0006475766 0.9979966
## Not Applicable 0.023676880 0.0013927577 0.0487465181 0.9261838# Quirky fix so that the files can be used again later
rm(mort)
rm(x)
rm(first_three)
rm(first_three_2)
gc()## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1886841 100.8 3205452 171.2 3205452 171.2
## Vcells 12926707 98.7 20761330 158.4 17964950 137.1Working with Web Data in R
Chapter 1 - Downloading Files and Using API Clients
Introduction: Working with Web Data in R:
- Methods for getting data from the internet in to R - frequently automatic, such as giving an internet address to read.csv()
- Using the httr package (tidyverse) to query API using GET() and POST()
- Using JSON and XML formats (nested data structures)
- CSS (cascading style sheets) for extracts
- Can use download.file() so that there is no need for repeatedly querying the same remote files
- You could use write.table(), but then you have to worry about accidentally writing out data in a format R can’t read back in
- An easy way to avoid this risk is to use saveRDS() and readRDS(), which save R objects in an R-specific file format, with the data structure intact
- That means you can use it for any type of R object (even ones that don’t turn into tables easily), and not worry you’ll lose data reading it back in
- saveRDS() takes two arguments, object, pointing to the R object to save and file pointing to where to save it to
- readRDS() expects file, referring to the path to the RDS file to read in
- An easy way to avoid this risk is to use saveRDS() and readRDS(), which save R objects in an R-specific file format, with the data structure intact
Understanding Application Programming Interfaces (API) - automatically handling data changes:
- Data are frequently made available by way of API
- “websites, but for machines”, allowing you to query/download data automatically
- “websites, but for machines”, allowing you to query/download data automatically
- R has several API interaction capabilities
- Native interfaces to API
- Hides API complexity
- Allows for reading data as R object
- Native interfaces to API
- Can find R packages for API by googling CRAN - packages frequently exist already
- Example is library(pageviews) to get pageview counts
Access tokens and API:
- API cients (by way of R packages) abstract away the complications of getting the data
- The API owner frequently does care how your API client interacts with it, though
- Overwhelming API causes problems for owner and many users
- Access tokens are sometimes used to monitor and throttle usage
- Overwhelming API causes problems for owner and many users
- Getting access tokens is frequently straightforward
- Usually requires registering an e-mail address
- Sometimes requires an explanation
- Example is www.wordnik.com, which can be accessed by way of library(bidnik)
- Usually requires registering an e-mail address
Example code includes:
# Here are the URLs! As you can see they're just normal strings
csv_url <- "http://s3.amazonaws.com/assets.datacamp.com/production/course_1561/datasets/chickwts.csv"
tsv_url <- "http://s3.amazonaws.com/assets.datacamp.com/production/course_3026/datasets/tsv_data.tsv"
# Read a file in from the CSV URL and assign it to csv_data
csv_data <- read.csv(csv_url)
# Read a file in from the TSV URL and assign it to tsv_data
tsv_data <- read.delim(tsv_url)
# Examine the objects with head()
head(csv_data)## weight feed
## 1 179 horsebean
## 2 160 horsebean
## 3 136 horsebean
## 4 227 horsebean
## 5 217 horsebean
## 6 168 horsebeanhead(tsv_data)## weight feed
## 1 179 horsebean
## 2 160 horsebean
## 3 136 horsebean
## 4 227 horsebean
## 5 217 horsebean
## 6 168 horsebean# Download the file with download.file()
download.file(url = csv_url, destfile = "./RInputFiles/feed_data.csv")
# Read it in with read.csv()
csv_data <- read.csv("./RInputFiles/feed_data.csv")
# Add a new column: square_weight
csv_data$square_weight <- csv_data$weight ** 2
# Save it to disk with saveRDS()
saveRDS(csv_data, "./RInputFiles/modified_feed_data.RDS")
# Read it back in with readRDS()
modified_feed_data <- readRDS("./RInputFiles/modified_feed_data.RDS")
# Examine modified_feed_data
str(modified_feed_data)## 'data.frame': 71 obs. of 3 variables:
## $ weight : int 179 160 136 227 217 168 108 124 143 140 ...
## $ feed : Factor w/ 6 levels "casein","horsebean",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ square_weight: num 32041 25600 18496 51529 47089 ...# Load pageviews
# library(pageviews)
# Get the pageviews for "Hadley Wickham"
hadley_pageviews <- pageviews::article_pageviews(project = "en.wikipedia", "Hadley Wickham")
# Examine the resulting object
str(hadley_pageviews)## 'data.frame': 1 obs. of 8 variables:
## $ project : chr "wikipedia"
## $ language : chr "en"
## $ article : chr "Hadley_Wickham"
## $ access : chr "all-access"
## $ agent : chr "all-agents"
## $ granularity: chr "daily"
## $ date : POSIXct, format: "2015-10-01"
## $ views : num 53# Load birdnik
# library(birdnik)
# Get the word frequency for "vector", using api_key to access it
# vector_frequency <- word_frequency(api_key, "vector")Chapter 2 - Using httr to interact with API Directly
GET and POST requests in theory - https and web requests in theory:
- Interactions on the internet can be though of as the client-server communication
- The most common request is “GET”, which is the client request for something from the server
- The parallel is “POST”, which is asking the server to accept something from the client
- HEAD is similar to head()
- DELETE is a request to the server to get rid of something - typically not needed
- The parallel is “POST”, which is asking the server to accept something from the client
- The httr package enables basic communication in R
- response <- httr::GET(url=) # will get that url
- httr::content(response) # will tell you about the response
- response <- httr::POST(url=) is for posting, and the recipient can figure out what to do with the data
- response <- httr::GET(url=) # will get that url
Graceful httr - code that responds appropriately and constructs its own url:
- Error handling - all httr requests come back with an error code (status)
- Status: 200 (completed) - starts with 2/3 is usually fine
- Status: 404 (no clue where to look) - starts with 4 is usually error in your code
- Status: starts with 5 is usually error in their code
- Can check for bad codes with http_error()
- Status: 200 (completed) - starts with 2/3 is usually fine
- URL construction - frequently most of the text stays the same, with just the occasional change in other components that do
- Directory based url are based on / and can be created using paste(sep=“/”) - very common, and very easy to create
- Parameter based url use text like https://fakeurl.com/api.php?a=1&b=2 and can be created using GET() with its named list of parameters
- GET(“fakeurl.com/api.php”, query = list(fruit = “peaches”, day = “thursday”))
- Directory based url are based on / and can be created using paste(sep=“/”) - very common, and very easy to create
Respectful API Usage - usage that works for the API owners as well as the clients:
- User agents - bits of text that ID your browser, give the server some idea of what you are trying to do, can be set with user_agent(), add an e-mail address, etc.
- Many API have rate-limiter capability - exceed and you will be blocked
- Keep an interval between requests, such as having a sleep (or similar) capability between requests using Sys.sleep()
Example code includes:
# Load the httr package
library(httr)
# Make a GET request to http://httpbin.org/get
get_result <- GET("http://httpbin.org/get")
# Print it to inspect it
# get_result
# Make a POST request to http://httpbin.org/post with the body "this is a test"
# post_result <- POST(url="http://httpbin.org/post", body="this is a test")
# Print it to inspect it
# post_result
url <- "https://wikimedia.org/api/rest_v1/metrics/pageviews/per-article/en.wikipedia.org/all-access/all-agents/Hadley_Wickham/daily/20170101/20170102"
# Make a GET request to url and save the results
pageview_response <- GET(url)
# Call content() to retrieve the data the server sent back
pageview_data <- content(pageview_response)
# Examine the results with str()
str(pageview_data)## List of 1
## $ items:List of 2
## ..$ :List of 7
## .. ..$ project : chr "en.wikipedia"
## .. ..$ article : chr "Hadley_Wickham"
## .. ..$ granularity: chr "daily"
## .. ..$ timestamp : chr "2017010100"
## .. ..$ access : chr "all-access"
## .. ..$ agent : chr "all-agents"
## .. ..$ views : int 45
## ..$ :List of 7
## .. ..$ project : chr "en.wikipedia"
## .. ..$ article : chr "Hadley_Wickham"
## .. ..$ granularity: chr "daily"
## .. ..$ timestamp : chr "2017010200"
## .. ..$ access : chr "all-access"
## .. ..$ agent : chr "all-agents"
## .. ..$ views : int 86fake_url <- "http://google.com/fakepagethatdoesnotexist"
# Make the GET request
request_result <- GET(fake_url)
# Check request_result
if(http_error(request_result)){
warning("The request failed")
} else {
content(request_result)
}## Warning: The request failed# Construct a directory-based API URL to `http://swapi.co/api`,
# looking for person `1` in `people`
directory_url <- paste("http://swapi.co/api", "people", 1, sep = "/")
# Make a GET call with it
result <- GET(directory_url)
# Create list with nationality and country elements
query_params <- list(nationality = "americans",
country = "antigua")
# Make parameter-based call to httpbin, with query_params
parameter_response <- GET("https://httpbin.org/get", query = query_params)
# Print parameter_response
parameter_response## Response [https://httpbin.org/get?nationality=americans&country=antigua]
## Date: 2018-02-16 13:02
## Status: 200
## Content-Type: application/json
## Size: 425 B
## {
## "args": {
## "country": "antigua",
## "nationality": "americans"
## },
## "headers": {
## "Accept": "application/json, text/xml, application/xml, */*",
## "Accept-Encoding": "gzip, deflate",
## "Connection": "close",
## "Host": "httpbin.org",
## ...# Do not change the url
# url <- "https://wikimedia.org/api/rest_v1/metrics/pageviews/per-article/en.wikipedia/all-access/all-agents/Aaron_Halfaker/daily/2015100100/2015103100"
# Add the email address and the test sentence inside user_agent()
# server_response <- GET(url, user_agent("my@email.address this is a test"))
# Construct a vector of 2 URLs
urls <- c("http://fakeurl.com/api/1.0/", "http://fakeurl.com/api/2.0/")
for(url in urls){
# Send a GET request to url
result <- GET(url)
# Delay for 5 seconds between requests
Sys.sleep(1)
}
get_pageviews <- function(article_title){
url <- paste0("https://wikimedia.org/api/rest_v1/metrics/pageviews/per-article/en.wikipedia/all-access/all-agents", article_title, "daily/2015100100/2015103100", sep = "/")
response <- GET(url, user_agent("my@email.com this is a test"))
if(http_error(response)){
stop("the request failed" )
} else {
result <- content(response)
return(result)
}
}Chapter 3 - Handling JSON and XML
JSON is a dictionary-like format (plain text) foe sending data on the internet:
- All JSON structures are made up of objects (name-value pairs in parentheses) {“a” : “b” , “c” : “d”} and arrays [1977, 1980]
- Values can be “string”, number, true, false, null, another object or array
- Complicated hierarchy can easily be represented
- Values can be “string”, number, true, false, null, another object or array
- Can find the type of data using httr::http_type(response)
Manipulating JSON - lists are the natural R hierarchy for JSON:
- fromJSON() will return named lists (if key-value pairs) and unnamed lists (if arrays)
- The simplifyDataFrame = TRUE argument will pull everything together in to a data frame if possible
- Alternately, can run lapply (or similar) over the list that has been returned
- The simplifyDataFrame = TRUE argument will pull everything together in to a data frame if possible
- One way to extract relevant data from that list is to use a package specifically designed for manipulating lists, rlist
- rlist provides two particularly useful functions for selecting and combining elements from a list: list.select() and list.stack()
- list.select() extracts sub-elements by name from each element in a list
- For example using the parsed movies data from the video (movies_list), we might ask for the title and year elements from each element: list.select(movies_list, title, year)
- The result is still a list, that is where list.stack() comes in. It will stack the elements of a list into a data frame: list.stack(list.select(movies_list, title, year))
- rlist provides two particularly useful functions for selecting and combining elements from a list: list.select() and list.stack()
XML Structure - plain text like JSON, but with a very different structure:
- Consists of markup (tags) and struture (data)
- Tags begin with < and end with >
- Typically
some stuff
- Can privide attributes inside of tags, such as
more stuff
- There is no formal standard, though attributes are usually used only for metadata
- Tags begin with < and end with >
- XML is a hierarchical structure, and includes everything between the start tag and the end tag
- Each element can contain many other elements
- Sub-elements are considered to be “children” of the “parent” element they are part of; “children” of the same “parent” are called “sibling” tags
- Each element can contain many other elements
- Just like JSON, you should first verify the response is indeed XML with http_type() and by examining the result of content(r, as = “text”)
- Then you can turn the response into an XML document object with read_xml()
- One benefit of using the XML document object is the available functions that help you explore and manipulate the document
- For example xml_structure() will print a representation of the XML document that emphasizes the hierarchical structure by displaying the elements without the data
- Then you can turn the response into an XML document object with read_xml()
XPATH - language for specifying nodes in an XML document:
- XPATH looks a lot like file.path, since it uses forward slash / to find the requested sub-nodes
- xml_find_all(x=, xpath=) # x is the object such as movies_xml and path is the xpath such as “/movies/movie/title”; will return a “node set”
- xml_text() run on a “node set” will return the data in an easier to digest format
- The // means “any node at any level below”, so “//title” will grab any node, from any path, that is tagged as “title”
- The @ means to extract an attribute; so, //movie/@episode will create a node set of the episodes under the movie tags
- xml_text() run on a “node set” will return the data in an easier to digest format
- Alternate ways to extract attributes include xml_attr() and xml_attrs()
- xml_attrs() takes a nodeset and returns all of the attributes for every node in the nodeset
- xml_attr() takes a nodeset and an additional argument attr to extract a single named argument from each node in the nodeset
- xml_attrs() takes a nodeset and returns all of the attributes for every node in the nodeset
Example code includes:
rev_history <- function(title, format = "json"){
if (title != "Hadley Wickham") {
stop('rev_history() only works for `title = "Hadley Wickham"`')
}
if (format == "json"){
resp <- readRDS("had_rev_json.rds")
} else if (format == "xml"){
resp <- readRDS("had_rev_xml.rds")
} else {
stop('Invalid format supplied, try "json" or "xml"')
}
resp
}
test_json <- "{\"continue\":{\"rvcontinue\":\"20150528042700|664370232\",\"continue\":\"||\"},\"query\":{\"pages\":{\"41916270\":{\"pageid\":41916270,\"ns\":0,\"title\":\"Hadley Wickham\",\"revisions\":[{\"user\":\"214.28.226.251\",\"anon\":\"\",\"timestamp\":\"2015-01-14T17:12:45Z\",\"comment\":\"\",\"contentformat\":\"text/x-wiki\",\"contentmodel\":\"wikitext\",\"*\":\"'''Hadley Mary Helen Wickham III''' is a [[statistician]] from [[New Zealand]] who is currently Chief Scientist at [[RStudio]]<ref>{{cite web|url=http://washstat.org/wss1310.shtml |title=Washington Statistical Society October 2013 Newsletter |publisher=Washstat.org |date= |accessdate=2014-02-12}}</ref><ref>{{cite web|url=http://news.idg.no/cw/art.cfm?id=F66B12BB-D13E-94B0-DAA22F5AB01BEFE7 |title=60+ R resources to improve your data skills ( - Software ) |publisher=News.idg.no |date= |accessdate=2014-02-12}}</ref> and an [[Professors_in_the_United_States#Adjunct_professor|adjunct]] [[Assistant Professor]] of statistics at [[Rice University]].<ref name=\\\"about\\\">{{cite web|url=http://www.rstudio.com/about/ |title=About - RStudio |accessdate=2014-08-13}}</ref> He is best known for his development of open-source statistical analysis software packages for [[R (programming language)]] that implement logics of [[data visualisation]] and data transformation. Wickham completed his undergraduate studies at the [[University of Auckland]] and his PhD at [[Iowa State University]] under the supervision of Di Cook and Heike Hoffman.<ref>{{cite web|URL=http://blog.revolutionanalytics.com/2010/09/the-r-files-hadley-wickham.html |title= The R-Files: Hadley Wickham}}</ref> In 2006 he was awarded the [[John_Chambers_(statistician)|John Chambers]] Award for Statistical Computing for his work developing tools for data reshaping and visualisation.<ref>{{cite web|url=http://stat-computing.org/awards/jmc/winners.html |title=John Chambers Award Past winners|publisher=ASA Sections on Statistical Computing, Statistical Graphics,|date= |accessdate=2014-08-12}}</ref>\\n\\nHe is a prominent and active member of the [[R (programming language)|R]] user community and has developed several notable and widely used packages including [[ggplot2]], plyr, dplyr, and reshape2.<ref name=\\\"about\\\" /><ref>{{cite web|url=http://www.r-statistics.com/2013/06/top-100-r-packages-for-2013-jan-may/ |title=Top 100 R Packages for 2013 (Jan-May)! |publisher=R-statistics blog |date= |accessdate=2014-08-12}}</ref>\"},{\"user\":\"73.183.151.193\",\"anon\":\"\",\"timestamp\":\"2015-01-15T15:49:34Z\",\"comment\":\"\",\"contentformat\":\"text/x-wiki\",\"contentmodel\":\"wikitext\",\"*\":\"'''Hadley Wickham''' is a [[statistician]] from [[New Zealand]] who is currently Chief Scientist at [[RStudio]]<ref>{{cite web|url=http://washstat.org/wss1310.shtml |title=Washington Statistical Society October 2013 Newsletter |publisher=Washstat.org |date= |accessdate=2014-02-12}}</ref><ref>{{cite web|url=http://news.idg.no/cw/art.cfm?id=F66B12BB-D13E-94B0-DAA22F5AB01BEFE7 |title=60+ R resources to improve your data skills ( - Software ) |publisher=News.idg.no |date= |accessdate=2014-02-12}}</ref> and an [[Professors_in_the_United_States#Adjunct_professor|adjunct]] [[Assistant Professor]] of statistics at [[Rice University]].<ref name=\\\"about\\\">{{cite web|url=http://www.rstudio.com/about/ |title=About - RStudio |accessdate=2014-08-13}}</ref> He is best known for his development of open-source statistical analysis software packages for [[R (programming language)]] that implement logics of [[data visualisation]] and data transformation. Wickham completed his undergraduate studies at the [[University of Auckland]] and his PhD at [[Iowa State University]] under the supervision of Di Cook and Heike Hoffman.<ref>{{cite web|URL=http://blog.revolutionanalytics.com/2010/09/the-r-files-hadley-wickham.html |title= The R-Files: Hadley Wickham}}</ref> In 2006 he was awarded the [[John_Chambers_(statistician)|John Chambers]] Award for Statistical Computing for his work developing tools for data reshaping and visualisation.<ref>{{cite web|url=http://stat-computing.org/awards/jmc/winners.html |title=John Chambers Award Past winners|publisher=ASA Sections on Statistical Computing, Statistical Graphics,|date= |accessdate=2014-08-12}}</ref>\\n\\nHe is a prominent and active member of the [[R (programming language)|R]] user community and has developed several notable and widely used packages including [[ggplot2]], plyr, dplyr, and reshape2.<ref name=\\\"about\\\" /><ref>{{cite web|url=http://www.r-statistics.com/2013/06/top-100-r-packages-for-2013-jan-may/ |title=Top 100 R Packages for 2013 (Jan-May)! |publisher=R-statistics blog |date= |accessdate=2014-08-12}}</ref>\"},{\"user\":\"FeanorStar7\",\"timestamp\":\"2015-01-24T16:34:31Z\",\"comment\":\"/* External links */ add LCCN and cats\",\"contentformat\":\"text/x-wiki\",\"contentmodel\":\"wikitext\",\"*\":\"'''Hadley Wickham''' is a [[statistician]] from [[New Zealand]] who is currently Chief Scientist at [[RStudio]]<ref>{{cite web|url=http://washstat.org/wss1310.shtml |title=Washington Statistical Society October 2013 Newsletter |publisher=Washstat.org |date= |accessdate=2014-02-12}}</ref><ref>{{cite web|url=http://news.idg.no/cw/art.cfm?id=F66B12BB-D13E-94B0-DAA22F5AB01BEFE7 |title=60+ R resources to improve your data skills ( - Software ) |publisher=News.idg.no |date= |accessdate=2014-02-12}}</ref> and an [[Professors_in_the_United_States#Adjunct_professor|adjunct]] [[Assistant Professor]] of statistics at [[Rice University]].<ref name=\\\"about\\\">{{cite web|url=http://www.rstudio.com/about/ |title=About - RStudio |accessdate=2014-08-13}}</ref> He is best known for his development of open-source statistical analysis software packages for [[R (programming language)]] that implement logics of [[data visualisation]] and data transformation. Wickham completed his undergraduate studies at the [[University of Auckland]] and his PhD at [[Iowa State University]] under the supervision of Di Cook and Heike Hoffman.<ref>{{cite web|URL=http://blog.revolutionanalytics.com/2010/09/the-r-files-hadley-wickham.html |title= The R-Files: Hadley Wickham}}</ref> In 2006 he was awarded the [[John_Chambers_(statistician)|John Chambers]] Award for Statistical Computing for his work developing tools for data reshaping and visualisation.<ref>{{cite web|url=http://stat-computing.org/awards/jmc/winners.html |title=John Chambers Award Past winners|publisher=ASA Sections on Statistical Computing, Statistical Graphics,|date= |accessdate=2014-08-12}}</ref>\\n\\nHe is a prominent and active member of the [[R (programming language)|R]] user community and has developed several notable and widely used packages including [[ggplot2]], plyr, dplyr, and reshape2.<ref name=\\\"about\\\" /><ref>{{cite web|url=http://www.r-statistics.com/2013/06/top-100-r-packages-for-2013-jan-may/ |title=Top 100 R Packages for 2013 (Jan-May)! |publisher=R-statistics blog |date= |accessdate=2014-08-12}}</ref>\"},{\"user\":\"KasparBot\",\"timestamp\":\"2015-04-26T19:18:17Z\",\"comment\":\"authority control moved to wikidata\",\"contentformat\":\"text/x-wiki\",\"contentmodel\":\"wikitext\",\"*\":\"'''Hadley Wickham''' is a [[statistician]] from [[New Zealand]] who is currently Chief Scientist at [[RStudio]]<ref>{{cite web|url=http://washstat.org/wss1310.shtml |title=Washington Statistical Society October 2013 Newsletter |publisher=Washstat.org |date= |accessdate=2014-02-12}}</ref><ref>{{cite web|url=http://news.idg.no/cw/art.cfm?id=F66B12BB-D13E-94B0-DAA22F5AB01BEFE7 |title=60+ R resources to improve your data skills ( - Software ) |publisher=News.idg.no |date= |accessdate=2014-02-12}}</ref> and an [[Professors_in_the_United_States#Adjunct_professor|adjunct]] [[Assistant Professor]] of statistics at [[Rice University]].<ref name=\\\"about\\\">{{cite web|url=http://www.rstudio.com/about/ |title=About - RStudio |accessdate=2014-08-13}}</ref> He is best known for his development of open-source statistical analysis software packages for [[R (programming language)]] that implement logics of [[data visualisation]] and data transformation. Wickham completed his undergraduate studies at the [[University of Auckland]] and his PhD at [[Iowa State University]] under the supervision of Di Cook and Heike Hoffman.<ref>{{cite web|URL=http://blog.revolutionanalytics.com/2010/09/the-r-files-hadley-wickham.html |title= The R-Files: Hadley Wickham}}</ref> In 2006 he was awarded the [[John_Chambers_(statistician)|John Chambers]] Award for Statistical Computing for his work developing tools for data reshaping and visualisation.<ref>{{cite web|url=http://stat-computing.org/awards/jmc/winners.html |title=John Chambers Award Past winners|publisher=ASA Sections on Statistical Computing, Statistical Graphics,|date= |accessdate=2014-08-12}}</ref>\\n\\nHe is a prominent and active member of the [[R (programming language)|R]] user community and has developed several notable and widely used packages including [[ggplot2]], plyr, dplyr, and reshape2.<ref name=\\\"about\\\" /><ref>{{cite web|url=http://www.r-statistics.com/2013/06/top-100-r-packages-for-2013-jan-may/ |title=Top 100 R Packages for 2013 (Jan-May)! |publisher=R-statistics blog |date= |accessdate=2014-08-12}}</ref>\"},{\"user\":\"Spkal\",\"timestamp\":\"2015-05-06T18:24:57Z\",\"comment\":\"/* Bibliography */ Added his new book, R Packages\",\"contentformat\":\"text/x-wiki\",\"contentmodel\":\"wikitext\",\"*\":\"'''Hadley Wickham''' is a [[statistician]] from [[New Zealand]] who is currently Chief Scientist at [[RStudio]]<ref>{{cite web|url=http://washstat.org/wss1310.shtml |title=Washington Statistical Society October 2013 Newsletter |publisher=Washstat.org |date= |accessdate=2014-02-12}}</ref><ref>{{cite web|url=http://news.idg.no/cw/art.cfm?id=F66B12BB-D13E-94B0-DAA22F5AB01BEFE7 |title=60+ R resources to improve your data skills ( - Software ) |publisher=News.idg.no |date= |accessdate=2014-02-12}}</ref> and an [[Professors_in_the_United_States#Adjunct_professor|adjunct]] [[Assistant Professor]] of statistics at [[Rice University]].<ref name=\\\"about\\\">{{cite web|url=http://www.rstudio.com/about/ |title=About - RStudio |accessdate=2014-08-13}}</ref> He is best known for his development of open-source statistical analysis software packages for [[R (programming language)]] that implement logics of [[data visualisation]] and data transformation. Wickham completed his undergraduate studies at the [[University of Auckland]] and his PhD at [[Iowa State University]] under the supervision of Di Cook and Heike Hoffman.<ref>{{cite web|URL=http://blog.revolutionanalytics.com/2010/09/the-r-files-hadley-wickham.html |title= The R-Files: Hadley Wickham}}</ref> In 2006 he was awarded the [[John_Chambers_(statistician)|John Chambers]] Award for Statistical Computing for his work developing tools for data reshaping and visualisation.<ref>{{cite web|url=http://stat-computing.org/awards/jmc/winners.html |title=John Chambers Award Past winners|publisher=ASA Sections on Statistical Computing, Statistical Graphics,|date= |accessdate=2014-08-12}}</ref>\\n\\nHe is a prominent and active member of the [[R (programming language)|R]] user community and has developed several notable and widely used packages including [[ggplot2]], plyr, dplyr, and reshape2.<ref name=\\\"about\\\" /><ref>{{cite web|url=http://www.r-statistics.com/2013/06/top-100-r-packages-for-2013-jan-may/ |title=Top 100 R Packages for 2013 (Jan-May)! |publisher=R-statistics blog |date= |accessdate=2014-08-12}}</ref>\"}]}}}}"
# Get revision history for "Hadley Wickham"
resp_json <- rev_history("Hadley Wickham")
# Check http_type() of resp_json
http_type(resp_json)
# Examine returned text with content()
content(resp_json, as="text")
# Parse response with content()
content(resp_json, as="parsed")
# Parse returned text with fromJSON()
library(jsonlite)
fromJSON(content(resp_json, as="text"))
# Load rlist
library(rlist)
# Examine output of this code
str(content(resp_json), max.level = 4)
# Store revision list
revs <- content(resp_json)$query$pages$`41916270`$revisions
# Extract the user element
user_time <- list.select(revs, user, timestamp)
# Print user_time
user_time
# Stack to turn into a data frame
list.stack(user_time)
# Load dplyr
library(dplyr)
# Pull out revision list
revs <- content(resp_json)$query$pages$`41916270`$revisions
# Extract user and timestamp
revs %>%
bind_rows() %>%
select(user, timestamp)
# Load xml2
library(xml2)
# Get XML revision history
resp_xml <- rev_history("Hadley Wickham", format = "xml")
# Check response is XML
http_type(resp_xml)
# Examine returned text with content()
rev_text <- content(resp_xml, as="text")
rev_text
# Turn rev_text into an XML document
rev_xml <- read_xml(rev_text)
# Examine the structure of rev_xml
str(rev_xml)
# Load xml2
library(xml2)
# Get XML revision history
resp_xml <- rev_history("Hadley Wickham", format = "xml")
# Check response is XML
http_type(resp_xml)
# Examine returned text with content()
rev_text <- content(resp_xml, as="text")
rev_text
# Turn rev_text into an XML document
rev_xml <- read_xml(rev_text)
# Examine the structure of rev_xml
xml_structure(rev_xml)
# Find all nodes using XPATH "/api/query/pages/page/revisions/rev"
xml_find_all(rev_xml, "/api/query/pages/page/revisions/rev")
# Find all rev nodes anywhere in document
rev_nodes <- xml_find_all(rev_xml, "//rev")
# Use xml_text() to get text from rev_nodes
xml_text(rev_nodes)
# All rev nodes
rev_nodes <- xml_find_all(rev_xml, "//rev")
# The first rev node
first_rev_node <- xml_find_first(rev_xml, "//rev")
# Find all attributes with xml_attrs()
xml_attrs(first_rev_node)
# Find user attribute with xml_attr()
xml_attr(first_rev_node, attr="user")
# Find user attribute for all rev nodes
xml_attr(rev_nodes, attr="user")
# Find anon attribute for all rev nodes
xml_attr(rev_nodes, attr="anon")
get_revision_history <- function(article_title){
# Get raw revision response
rev_resp <- rev_history(article_title, format = "xml")
# Turn the content() of rev_resp into XML
rev_xml <- read_xml(content(rev_resp, "text"))
# Find revision nodes
rev_nodes <- xml_find_all(rev_xml, "//rev")
# Parse out usernames
user <- xml_attr(rev_nodes, attr="user")
# Parse out timestamps
timestamp <- readr::parse_datetime(xml_attr(rev_nodes, "timestamp"))
# Parse out content
content <- xml_text(rev_nodes)
# Return data frame
data.frame(user = user,
timestamp = timestamp,
content = substr(content, 1, 40))
}
# Call function for "Hadley Wickham"
get_revision_history(article_title = "Hadley Wickham")Chapter 4 - Web Scraping with XPATH
Web scraping 101 - sometimes a website does not have an API, so a different approach is required:
- Web scraping is the process of grabbling the full html and then parsing the data as needed
- The “selector” plug-in for a browser can be helpful for finding IDs associated with examples of interest
- There is a package “rvest” that helps to simplify the process of web scraping
- rvest::read_html(url=) # returns an XML document
- html_node() will extract contents with XPATH (???) - the argument to html_node should be the returned XML document from the previous step
- rvest::read_html(url=) # returns an XML document
HTML structure - basically, content within tags, much like XML:
- For example
This is a test
requests that “This is a test” be available in paragraph form
- Attributes can be stored also, such as this is a test
- Parameters can incorporate formatting, style, and the like
- The rvest package has the means for extracting the data from html
- html_text(x=) for text contents
- html_attr(x=, name=) to get a specific attribute
- html_name(x=) to get the tag name
- html_text(x=) for text contents
Reformatting data (especially to a rectangular format such as a data frame):
- Turning html tables (tables are a structure in html) in to data frames
- They can be identified in raw html from
- They can be turned in to tables using html_table()
- Can assign column names using colnames() as per normal R
- They can be identified in raw html from
- Turning html non-tables in to data frames
- Use data.frame() with the vectors of text or names or attributes or the like
Example code includes:
# Load rvest
library(rvest)## Loading required package: xml2##
## Attaching package: 'rvest'## The following object is masked from 'package:readr':
##
## guess_encoding# Hadley Wickham's Wikipedia page
test_url <- "https://en.wikipedia.org/wiki/Hadley_Wickham"
# Read the URL stored as "test_url" with read_html()
test_xml <- read_html(test_url)
# Print test_xml
test_xml## {xml_document}
## <html class="client-nojs" lang="en" dir="ltr">
## [1] <head>\n<meta http-equiv="Content-Type" content="text/html; charset= ...
## [2] <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-sub ...test_node_xpath <- "//*[contains(concat( \" \", @class, \" \" ), concat( \" \", \"vcard\", \" \" ))]"
# Use html_node() to grab the node with the XPATH stored as `test_node_xpath`
node <- html_node(x = test_xml, xpath = test_node_xpath)
# Print the first element of the result
node[1]## $node
## <pointer: 0x000000000b91bb80># The first thing we'll grab is a name, from the first element of the previously extracted table (now stored as table_element)
table_element <- node
# Extract the name of table_element
element_name <- html_name(table_element)
# Print the name
element_name## [1] "table"second_xpath_val <- "//*[contains(concat( \" \", @class, \" \" ), concat( \" \", \"fn\", \" \" ))]"
# Extract the element of table_element referred to by second_xpath_val and store it as page_name
page_name <- html_node(x = table_element, xpath = second_xpath_val)
# Extract the text from page_name
page_title <- html_text(page_name)
# Print page_title
page_title## [1] "Hadley Wickham"# Turn table_element into a data frame and assign it to wiki_table
wiki_table <- html_table(table_element)
# Print wiki_table
wiki_table## Hadley Wickham
## 1
## 2 Residence
## 3 Alma mater
## 4 Known for
## 5 Awards
## 6 Scientific career
## 7 Fields
## 8 Thesis
## 9 Doctoral advisors
## 10 Doctoral students
## 11
## Hadley Wickham
## 1
## 2 United States
## 3 Iowa State University, University of Auckland
## 4 R programming language packages
## 5 John Chambers Award (2006)\nFellow of the American Statistical Association (2015)
## 6 Scientific career
## 7 Statistics\nData science\nR (programming language)
## 8 Practical tools for exploring data and models (2008)
## 9 Di Cook\nHeike Hofmann
## 10 Garrett Grolemund
## 11# Rename the columns of wiki_table
colnames(wiki_table) <- c("key", "value")
# Remove the empty row from wiki_table
cleaned_table <- subset(wiki_table, !(key == ""))
# Print cleaned_table
cleaned_table## key
## 2 Residence
## 3 Alma mater
## 4 Known for
## 5 Awards
## 6 Scientific career
## 7 Fields
## 8 Thesis
## 9 Doctoral advisors
## 10 Doctoral students
## value
## 2 United States
## 3 Iowa State University, University of Auckland
## 4 R programming language packages
## 5 John Chambers Award (2006)\nFellow of the American Statistical Association (2015)
## 6 Scientific career
## 7 Statistics\nData science\nR (programming language)
## 8 Practical tools for exploring data and models (2008)
## 9 Di Cook\nHeike Hofmann
## 10 Garrett GrolemundChapter 5 - CSS Web Scraping and Final Case Study
CSS (cascading style sheets) web scraping in theory:
- CSS is for style, formatting, and the like
- Groups of CSS commands are associated to a class, allowing the class to be used in multiple areas
- .class_a { color: black; }
- .class_b { color: red; }
- Specific html can then be addressed using This is black
- .class_a { color: black; }
- CSS scraping is the concept of finding the class groups
- Works much like XPATH but will often grab many items rather than just a single element
- It’s more common with CSS selectors to use html_nodes()
- To select elements with a certain class, you add a . in front of the class name
- If you need to select an element based on its id, you add a # in front of the id name
- For example if this element was inside your HTML document:
- Introduction
- You could select it by its class using the CSS selector “.heading”, or by its id using the CSS selector “#intro”
- Works much like XPATH but will often grab many items rather than just a single element
Final case study: Introduction:
- Extracting an infobox from a Wikipedia page
- Get XML by way of API
- Extract infobox from the page
- Clean up and convert to data frame
- Wrap in a function for reproducibility
- Get XML by way of API
Wrap up:
- Downloading and reading flat files
- Designing and using API clients
- Web scraping using XPATHs and CSS
Example code includes:
library(rvest)## Loading required package: xml2##
## Attaching package: 'rvest'## The following object is masked from 'package:readr':
##
## guess_encoding# Hadley Wickham's Wikipedia page
test_url <- "https://en.wikipedia.org/wiki/Hadley_Wickham"
# Read the URL stored as "test_url" with read_html()
test_xml <- read_html(test_url)
# Print test_xml
test_xml## {xml_document}
## <html class="client-nojs" lang="en" dir="ltr">
## [1] <head>\n<meta http-equiv="Content-Type" content="text/html; charset= ...
## [2] <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-sub ...# Select the table elements
html_nodes(test_xml, css = "table")## {xml_nodeset (2)}
## [1] <table class="infobox biography vcard" style="width:22em">\n<tr>\n<t ...
## [2] <table class="nowraplinks hlist navbox-inner" style="border-spacing: ...# Select elements with class = "infobox"
html_nodes(test_xml, css = ".infobox")## {xml_nodeset (1)}
## [1] <table class="infobox biography vcard" style="width:22em">\n<tr>\n<t ...# Select elements with id = "firstHeading"
html_nodes(test_xml, css = "#firstHeading")## {xml_nodeset (1)}
## [1] <h1 id="firstHeading" class="firstHeading" lang="en">Hadley Wickham< ...# Extract element with class infobox
infobox_element <- html_nodes(test_xml, css = ".infobox")
# Get tag name of infobox_element
element_name <- html_name(infobox_element)
# Print element_name
element_name## [1] "table"# Extract element with class fn
page_name <- html_node(x = infobox_element, css=".fn")
# Get contents of page_name
page_title <- html_text(page_name)
# Print page_title
page_title## [1] "Hadley Wickham"# Load httr
library(httr)
# The API url
base_url <- "https://en.wikipedia.org/w/api.php"
# Set query parameters
query_params <- list(action="parse",
page="Hadley Wickham",
format="xml")
# Get data from API
resp <- GET(url = "https://en.wikipedia.org/w/api.php", query = query_params)
# Parse response
resp_xml <- content(resp)
# Load rvest
library(rvest)
# Read page contents as HTML
page_html <- read_html(xml_text(resp_xml))
# Extract infobox element
infobox_element <- html_node(page_html, css=".infobox")
# Extract page name element from infobox
page_name <- html_node(infobox_element, css=".fn")
# Extract page name as text
page_title <- html_text(page_name)
# Your code from earlier exercises
wiki_table <- html_table(infobox_element)
colnames(wiki_table) <- c("key", "value")
cleaned_table <- subset(wiki_table, !key == "")
# Create a dataframe for full name
name_df <- data.frame(key = "Full name", value = page_title)
# Combine name_df with cleaned_table
wiki_table2 <- rbind(name_df, cleaned_table)
# Print wiki_table
wiki_table2## key
## 1 Full name
## 2 Residence
## 3 Alma mater
## 4 Known for
## 5 Awards
## 6 Scientific career
## 7 Fields
## 8 Thesis
## 9 Doctoral advisors
## 10 Doctoral students
## value
## 1 Hadley Wickham
## 2 United States
## 3 Iowa State University, University of Auckland
## 4 R programming language packages
## 5 John Chambers Award (2006)\nFellow of the American Statistical Association (2015)
## 6 Scientific career
## 7 Statistics\nData science\nR (programming language)
## 8 Practical tools for exploring data and models (2008)
## 9 Di Cook\nHeike Hofmann
## 10 Garrett Grolemundlibrary(httr)
library(rvest)
library(xml2)
get_infobox <- function(title){
base_url <- "https://en.wikipedia.org/w/api.php"
# Change "Hadley Wickham" to title
query_params <- list(action = "parse",
page = title,
format = "xml")
resp <- GET(url = base_url, query = query_params)
resp_xml <- content(resp)
page_html <- read_html(xml_text(resp_xml))
infobox_element <- html_node(x = page_html, css =".infobox")
page_name <- html_node(x = infobox_element, css = ".fn")
page_title <- html_text(page_name)
wiki_table <- html_table(infobox_element)
colnames(wiki_table) <- c("key", "value")
cleaned_table <- subset(wiki_table, !wiki_table$key == "")
name_df <- data.frame(key = "Full name", value = page_title)
wiki_table <- rbind(name_df, cleaned_table)
wiki_table
}
# Test get_infobox with "Hadley Wickham"
get_infobox(title = "Hadley Wickham")## key
## 1 Full name
## 2 Residence
## 3 Alma mater
## 4 Known for
## 5 Awards
## 6 Scientific career
## 7 Fields
## 8 Thesis
## 9 Doctoral advisors
## 10 Doctoral students
## value
## 1 Hadley Wickham
## 2 United States
## 3 Iowa State University, University of Auckland
## 4 R programming language packages
## 5 John Chambers Award (2006)\nFellow of the American Statistical Association (2015)
## 6 Scientific career
## 7 Statistics\nData science\nR (programming language)
## 8 Practical tools for exploring data and models (2008)
## 9 Di Cook\nHeike Hofmann
## 10 Garrett Grolemund# Try get_infobox with "Ross Ihaka"
get_infobox(title = "Ross Ihaka")## key
## 1 Full name
## 2 Ihaka at the 2010 New Zealand Open Source Awards
## 3 Residence
## 4 Alma mater
## 5 Known for
## 6 Awards
## 7 Scientific career
## 8 Fields
## 9 Institutions
## 10 Thesis
## 11 Doctoral advisor
## value
## 1 Ross Ihaka
## 2 Ihaka at the 2010 New Zealand Open Source Awards
## 3 Auckland, New Zealand
## 4 University of AucklandUniversity of California, Berkeley
## 5 R programming language
## 6 Pickering Medal (2008)
## 7 Scientific career
## 8 Statistical Computing
## 9 University of Auckland
## 10 Ruaumoko (1985)
## 11 David R. Brillinger# Try get_infobox with "Grace Hopper"
get_infobox(title = "Grace Hopper")## key
## 1 Full name
## 2 Rear Admiral Grace M. Hopper, 1984
## 3 Nickname(s)
## 4 Born
## 5 Died
## 6 Place of burial
## 7 Allegiance
## 8 Service/branch
## 9 Years of service
## 10 Rank
## 11 Awards
## value
## 1 Grace Murray Hopper
## 2 Rear Admiral Grace M. Hopper, 1984
## 3 "Amazing Grace"
## 4 (1906-12-09)December 9, 1906\nNew York City, New York, U.S.
## 5 January 1, 1992(1992-01-01) (aged 85)Arlington, Virginia, U.S.
## 6 Arlington National Cemetery
## 7 United States of America
## 8 United States Navy
## 9 1943–1966, 1967–1971, 1972–1986
## 10 Rear admiral (lower half)
## 11 Defense Distinguished Service Medal Legion of Merit Meritorious Service Medal American Campaign Medal World War II Victory Medal National Defense Service Medal Armed Forces Reserve Medal with two Hourglass Devices Naval Reserve Medal Presidential Medal of Freedom (posthumous)Data Visualization in R with lattice
Chapter 1 - Basic plotting with lattice
Introduction - general objectives:
- Visualization may be for EDA or for reporting results
- Three basic graphing capabilities in R
- Base - powerful but not flexible
- lattice - based on “Trellis graphics” (Cleveland)
- ggplot2 - based on “Grammar of Graphics” (Wilkinson)
- Base - powerful but not flexible
- This course will cover lattice graphics for both EDA and reporting
- Focus will be on the USCancerRates dataset, with exploration of variance by gender and location
- histogram(~ x, data=) # lattice for make a histogram (default appears to be RELATIVE frequency by bin)
- xyplot(y ~ x, data=) # lattice for make an xy plot
- The modeling calls are similar to what would be seen in an lm()
- histogram(~ x, data=) # lattice for make a histogram (default appears to be RELATIVE frequency by bin)
Optional arguments:
- Plotting functions in lattice frequently require two arguments - formula and data set
- Additional options are available and can be supplied to certain functions
- For example, histogram(~ x, data=, main=, xlab=) # will give the plot title “main” and the X-axis label “xlab”
- xyplot can also have a ylab=
- histogram can also have nint= (specifies the number of bins)
- The grid= argument of xyplot adds a background grid, while abline= adds a line with slope and intercept as specified
- For example, histogram(~ x, data=, main=, xlab=) # will give the plot title “main” and the X-axis label “xlab”
- In the case of histogram(), the optional argument type controls what is plotted on the y-axis. It can take three values:
- “percent”, the default, gives percentage or relative frequency
- “count” gives bin count, which is the default in hist()
- “density” gives a density histogram
- “percent”, the default, gives percentage or relative frequency
- The lattice function densityplot() creates kernel density plots (formula interface is similar to that of histogram())
- the formula should be written as ~ x to plot the values of the x column along the x-axis, and the estimated density on the y-axis
- A useful optional argument for densityplot() is plot.points, which can take values
- TRUE, the default, to plot the data points along the x-axis in addition to the density
- FALSE to suppress plotting the data points
- “jitter”, to plot the points along the y-axis but with some random jittering in the y-direction so that overlapping points are easier to see
- the formula should be written as ~ x to plot the values of the x column along the x-axis, and the estimated density on the y-axis
Box and whisker plots and reordering elements:
- Box and whisker plots are formed using bwplot(~ x, data=)
- Can serve a similar purpose as a histogram or density plot, and the formula is correspondingly similar
- bwplot(y ~ x, data=) will make box plots for x, split by each level of y (which needs to be a factor/categorical)
- bwplot(y ~ x, data=) will make box plots for x, split by each level of y (which needs to be a factor/categorical)
- The function reorder(myFactor, myData, myFunction, … ) will reorder factor variables for plotting
- For example, reorder(state, rate.male, median, na.rm=TRUE) will order the factor variable state by median(rate.male) in that state
- For example, reorder(state, rate.male, median, na.rm=TRUE) will order the factor variable state by median(rate.male) in that state
- Your task for this exercise is to produce a box-and-whisker plot where the whiskers extend to the data extremes
- These calculations are controlled by the coef argument of the R helper function boxplot.stats()
- A positive value of coef makes the whiskers extend to no more than coef times the length of the box
- The value of coef = 0 makes the whiskers extend to the data extremes
- These calculations are controlled by the coef argument of the R helper function boxplot.stats()
Example code includes:
data(airquality)
str(airquality)## 'data.frame': 153 obs. of 6 variables:
## $ Ozone : int 41 36 12 18 NA 28 23 19 8 NA ...
## $ Solar.R: int 190 118 149 313 NA NA 299 99 19 194 ...
## $ Wind : num 7.4 8 12.6 11.5 14.3 14.9 8.6 13.8 20.1 8.6 ...
## $ Temp : int 67 72 74 62 56 66 65 59 61 69 ...
## $ Month : int 5 5 5 5 5 5 5 5 5 5 ...
## $ Day : int 1 2 3 4 5 6 7 8 9 10 ...# Load the lattice package
library(lattice)
# Create the histogram
histogram(~ Ozone, data = airquality)
# Create the histogram
histogram(~ Ozone, data = airquality,
# Specify number of bins
nint = 15,
# Specify quantity displayed on y-axis
type = "count")
# Create the scatter plot
xyplot(Ozone ~ Solar.R, data = airquality)
# Create scatterplot
xyplot(Ozone ~ Temp, data = airquality,
# Add main label
main = "Environmental conditions in New York City (1973)",
# Add axis labels
ylab = "Ozone (ppb)",
xlab = "Temperature (Fahrenheit)")
# Create a density plot
densityplot(~ Ozone, data = airquality,
# Choose how raw data is shown
plot.points = "jitter")
data(USCancerRates, package="latticeExtra")
str(USCancerRates)## 'data.frame': 3041 obs. of 8 variables:
## $ rate.male : num 364 346 341 336 330 ...
## $ LCL95.male : num 311 274 304 289 293 ...
## $ UCL95.male : num 423 431 381 389 371 ...
## $ rate.female : num 151 140 182 185 172 ...
## $ LCL95.female: num 124 103 161 157 151 ...
## $ UCL95.female: num 184 190 206 218 195 ...
## $ state : Factor w/ 49 levels "Alabama","Alaska",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ county :Class 'AsIs' chr [1:3041] "Pickens County" "Bullock County" "Russell County" "Barbour County" ...rn_USCR <- row.names(USCancerRates)
# Create reordered variable
library(dplyr)
USCancerRates <-
mutate(USCancerRates,
state.ordered = reorder(state, rate.female, median, na.rm = TRUE)
)
# Create box and whisker plot
bwplot(state.ordered ~ rate.female, data = USCancerRates)
# Create box and whisker plot
bwplot(state.ordered ~ rate.female, data = USCancerRates,
# Change whiskers extent
coef = 0)
Chapter 2 - Conditioning and the Formula Interface
Conditioning - identify sources of variability in the data by examining sub-groups:
- Small multiple design - conditioning/faceting approach
- The conditioning operator in lattice is the single-pipe (|)
- xyplot(y ~ x | c, data=) # co is the conditioning variable in this example
- Can use the conditioning operator in any function within the lattice framework
- xyplot(y ~ x | c, data=) # co is the conditioning variable in this example
- The plus (+) operator is another way to condition - means condition on more than one variable
- histogram(~ a + b, outer=TRUE, layout=c(1, 2), data=) will put a separate histogram for b below the separate histogram for a, keeping both on the same scale
- The outer command determines how to interpret a+b
- The layout=c(1, 2) means 1 column and 2 rows - general format is layout=c(ncol, nrow, npages)
- histogram(~ a + b, outer=TRUE, layout=c(1, 2), data=) will put a separate histogram for b below the separate histogram for a, keeping both on the same scale
- Since count-based functions tend to have higher variances associated to higher means, the log transform for these can be valuable
- lattice, unlike ggplot2, allows you to have data in a wide format
Data summary and transformation - grouping:
- Data summarization can be especially valuable for reporting
- For example, may want to summarize cancer rates by state (median county) rather than by county
- The tapply() function can be valuable for applying a function across a vector
- To get both genders on the same plot but in different colors, use xyplot(State ~ Rate, data=, grid=TRUE, groups=Gender) # will treat the Gender as a separate group with different color on the same plot
- For example, may want to summarize cancer rates by state (median county) rather than by county
- New concept: groups - interpreted as a factor that defines sub-groups
- xyplot() and densityplot() support this, while histogram() does not
- Using auto.key = TRUE will add a legend telling which colors are associated to which groups
- xyplot() and densityplot() support this, while histogram() does not
- For more detailed control, the auto.key argument can be a list with various sub-components, the most useful of which are
- space: which can be “left”, “right”, “top”, or “bottom”
- columns: specifies the number of columns in which to divide up the levels
- title: specifies a title for the legend
- space: which can be “left”, “right”, “top”, or “bottom”
Incorporating external data sources:
- Can potentially split panels in to multiple pages or place multiple plots in the same pane
- For eample, could aggregate states by region and report states in the same region together
- The layout argument inside a lattice plotting function calls for layout=c(ncol, nrow)
- The between argument inside a lattice plotting function calls for spacing - bewteen=list(y=c(0, 0, 1, 0, 0)) will put a space of 1 between the third and fourth items
- For eample, could aggregate states by region and report states in the same region together
- The outer=FALSE makes the conditioning variable in to a grouping variable - more effective visual with multiple plots together on the same pane
- In a conditioned lattice plot, the panels are by default drawn starting from the bottom-left position, going right and then up
- This is patterned on the Cartesian coordinate system where the x-axis increases to the right and the y-axis increases from bottom to top
- This is patterned on the Cartesian coordinate system where the x-axis increases to the right and the y-axis increases from bottom to top
- Often, want to change this so that the layout is similar to a matrix or table, where rows start at the top
- The layout of any conditioned lattice plot can be changed to follow this scheme by adding the optional argument as.table = TRUE
The trellis object - lattice creates trellis objects rather than directly creating plots (as in base R):
- Can run the class(), summary() and the like, with auto-print and/or print() making the plot visible
- If you have a trellis object, the update() command can be used to modify the object
- In particular, their dimnames() are used as strip labels
- In particular, their dimnames() are used as strip labels
- Can think of the trellis object as being like a matrix, so t(trellisObject) will flip the rows/columns
- Depending on the amount of space available, a conditioned plot may have too many combinations to be displayed effectively
- Such plots can be split into multiple pages using the layout argument
- But another convenient way to explore large lattice plots is to subset them like a matrix or array, using the [ indexing operator, to display only parts of the plot at a time
- Such plots can be split into multiple pages using the layout argument
Example code includes:
# The airquality dataset has been pre-loaded
str(airquality)## 'data.frame': 153 obs. of 6 variables:
## $ Ozone : int 41 36 12 18 NA 28 23 19 8 NA ...
## $ Solar.R: int 190 118 149 313 NA NA 299 99 19 194 ...
## $ Wind : num 7.4 8 12.6 11.5 14.3 14.9 8.6 13.8 20.1 8.6 ...
## $ Temp : int 67 72 74 62 56 66 65 59 61 69 ...
## $ Month : int 5 5 5 5 5 5 5 5 5 5 ...
## $ Day : int 1 2 3 4 5 6 7 8 9 10 ...# Create a histogram
histogram(~ Ozone | factor(Month),
data = airquality,
# Define the layout
layout=c(2, 3),
# Change the x-axis label
xlab="Ozone (ppb)")
# USCancerRates has been pre-loaded
str(USCancerRates)## 'data.frame': 3041 obs. of 9 variables:
## $ rate.male : num 364 346 341 336 330 ...
## $ LCL95.male : num 311 274 304 289 293 ...
## $ UCL95.male : num 423 431 381 389 371 ...
## $ rate.female : num 151 140 182 185 172 ...
## $ LCL95.female : num 124 103 161 157 151 ...
## $ UCL95.female : num 184 190 206 218 195 ...
## $ state : Factor w/ 49 levels "Alabama","Alaska",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ county :Class 'AsIs' chr [1:3041] "Pickens County" "Bullock County" "Russell County" "Barbour County" ...
## $ state.ordered: Factor w/ 49 levels "Utah","New Mexico",..: 25 25 25 25 25 25 25 25 25 25 ...
## ..- attr(*, "scores")= num [1:49(1d)] 166 166 145 169 159 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "Alabama" "Alaska" "Arizona" "Arkansas" ...# Create a density plot
densityplot(~ rate.male + rate.female,
data = USCancerRates,
outer = TRUE,
# Suppress data points
plot.points = FALSE,
# Add a reference line
ref=TRUE)
# Create a density plot
densityplot(~ rate.male + rate.female,
data = USCancerRates,
# Set value of 'outer'
outer=FALSE,
# Add x-axis label
xlab="Rate (per 100,000)",
# Add a legend
auto.key=TRUE,
plot.points = FALSE,
ref = TRUE)
xyplot(Ozone ~ Temp, airquality, groups = Month,
# Complete the legend spec
auto.key = list(space = "right",
title = "Month",
text = month.name[5:9]))
USCancerRates <- USCancerRates %>%
mutate(division=state.division[match(state, state.name)])
# Create 'division.ordered' by reordering levels
USCancerRates <-
mutate(USCancerRates,
division.ordered = reorder(division,
rate.male + rate.female,
mean, na.rm = TRUE))
# Create conditioned scatter plot
xyplot(rate.female ~ rate.male | division.ordered,
data = USCancerRates,
# Add reference grid
grid = TRUE,
# Add reference line
abline = c(0, 1))
# Levels of division.ordered
levels(USCancerRates$division.ordered)## [1] "Mountain" "West North Central" "Pacific"
## [4] "Middle Atlantic" "New England" "East North Central"
## [7] "West South Central" "South Atlantic" "East South Central"# Specify the as.table argument
xyplot(rate.female ~ rate.male | division.ordered,
data = USCancerRates,
grid = TRUE, abline = c(0, 1),
as.table=TRUE)
# Create box-and-whisker plot
bwplot(division.ordered ~ rate.male + rate.female,
data = USCancerRates,
outer = TRUE,
# Add a label for the x-axis
xlab="Rate (per 100,000)",
# Add strip labels
strip = strip.custom(factor.levels = c("Male", "Female")))
# Create "trellis" object
tplot <-
densityplot(~ rate.male + rate.female | division.ordered,
data = USCancerRates, outer = TRUE,
plot.points = FALSE, as.table = TRUE)
# Change names for the second dimension
dimnames(tplot)[[2]] <- c("Male", "Female")
# Update x-axis label and plot
update(tplot, xlab = "Rate")
# Create "trellis" object
tplot <-
densityplot(~ rate.male + rate.female | division.ordered,
data = USCancerRates, outer = TRUE,
plot.points = FALSE, as.table = TRUE)
# Inspect dimension
dim(tplot)## [1] 9 2dimnames(tplot)## $division.ordered
## [1] "Mountain" "West North Central" "Pacific"
## [4] "Middle Atlantic" "New England" "East North Central"
## [7] "West South Central" "South Atlantic" "East South Central"
##
## [[2]]
## [1] "rate.male" "rate.female"# Select subset retaining only last three divisions
tplot[7:9, ]
Chapter 3 - Controlling scales and graphical parameters
Combining scales:
- Can use dotplot(y ~ x | c + d, data=, as.table=TRUE) to have a conditioned dot-plot on c and d
- Expects a categorical variable on at least one of the axes (typically, but not always, y)
- Expects a categorical variable on at least one of the axes (typically, but not always, y)
- The default for axis limits is for them to be common across all the panels - typically, best for interpretation
- Can override the default behavior using the scales argument, a list with named components
- relation = “same” is the default
- relation = “free” allows independence for each panel
- relation = “sliced” allows different limits for each panel, but with same range (???)
- Can override the default behavior using the scales argument, a list with named components
- The call to scales is fairly complicated
- scales = list(x = list(relation = “free”)) # asks for an x-axis scale to be free
- scales = list(x = list(relation = “free”)) # asks for an x-axis scale to be free
- Some other useful sub-components of the scales argument are:
- tick.number: approximate number of tick marks / labels
- alternating: 1 puts labels on the left/bottom boundary, 2 on the right/top, and 3 on both sides. The value can be a vector, in which case it applies row-wise or column-wise
- rot: angle in degrees to rotate axis labels
- tick.number: approximate number of tick marks / labels
Logarithmic scales:
- Can use dotplot(y ~ x | c, data=, groups=d, as.table=TRUE) will use d as a grouping variable with the plots only conditioned on c
- Can use the log() transform directly on the y variable to help with visualizing the data
- Alternately, can keep the data the same but just stretch the scales
- dotplot(y ~ x | c, data=, groups=d, scales=list(x = list(log = 2, equispaced.log=FALSE)), auto.key=list(columns=2))
- dotplot(y ~ x | c, data=, groups=d, scales=list(x = list(log = 2, equispaced.log=FALSE)), auto.key=list(columns=2))
- There is one more component you need to know, equispaced.log
- This component indicates if the tick marks are equispaced when log scales are in use
- By default, equispaced.log is set to TRUE
- Note: If you set equispaced.log = FALSE, you don’t have to explicitly specify a base for the log component; just log = TRUE should do the trick!
- This component indicates if the tick marks are equispaced when log scales are in use
Graphical parameters:
- A collection of graphical parameters is referred to as a theme, frequently stored globally so it can be easily re-used
- The trellis.par.set(myTheme) will work to set myTheme as the theme for the upcoming plot
- The latticeExtra package has ggplot2like() which will help match up the ggplot2 defaults
- The latticeExtra package has ggplot2like() which will help match up the ggplot2 defaults
- Can also control graphical parameters by way of calls within a graphin function
- For example, pch=15, col=c(“red”, “blue”)
- For example, pch=15, col=c(“red”, “blue”)
- Changing the graphical theme using trellis.par.set(), as demonstrated in the preceding video, makes the changes permanent, applying to all subsequent plots, until the theme is reset
- If you wish to make changes for a specific plot, an easier alternative is to supply the theme as the optional argument par.settings to a high-level call
- In that case, the settings will be associated only to that particular call
- In this exercise, you will use this approach to create a dot plot of the WorldPhones data with the ggplot2like() theme
- As we saw earlier, changing the theme alone may be insufficient; we also need to change other things like the spacing between panels
- Such settings (which are not considered graphical parameters) can also be customized through a list of “options”
- To go with the ggplot2like() theme, the latticeExtra package also provides a suitable list of options, produced by ggplot2like.opts()
- If you wish to make changes for a specific plot, an easier alternative is to supply the theme as the optional argument par.settings to a high-level call
- Options can be associated to a particular plot by specifying it as the lattice.options argument in a high-level call, or set more permanently using the lattice.options() function
Using simpleTheme():
- Empty circles are the default plotting symbol
- The pch=16 will create filled-in circles
- The pch=16 will create filled-in circles
- Interesting, changing parameters like pch in the function call apply only to the data, not to the legend describing the data
- Can instead make changes that apply to everything by specifying (inside the function) par.settings = simpleTheme(pch=16, col=c(“red”, “blue”))
- The simpleTheme() call will only change the requested options, leaving the global theme for everything else
- Can instead make changes that apply to everything by specifying (inside the function) par.settings = simpleTheme(pch=16, col=c(“red”, “blue”))
Example code includes:
# The lattice package and the USMortality dataset have been pre-loaded.
Status <- factor(c('Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural', 'Urban', 'Rural'), levels=c("Rural", "Urban")
)
Sex <- factor(c('Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female', 'Male', 'Male', 'Female', 'Female'), levels=c("Female", "Male")
)
Cause <- factor(c('Heart disease', 'Heart disease', 'Heart disease', 'Heart disease', 'Cancer', 'Cancer', 'Cancer', 'Cancer', 'Lower respiratory', 'Lower respiratory', 'Lower respiratory', 'Lower respiratory', 'Unintentional injuries', 'Unintentional injuries', 'Unintentional injuries', 'Unintentional injuries', 'Cerebrovascular diseases', 'Cerebrovascular diseases', 'Cerebrovascular diseases', 'Cerebrovascular diseases', 'Alzheimers', 'Alzheimers', 'Alzheimers', 'Alzheimers', 'Diabetes', 'Diabetes', 'Diabetes', 'Diabetes', 'Flu and pneumonia', 'Flu and pneumonia', 'Flu and pneumonia', 'Flu and pneumonia', 'Suicide', 'Suicide', 'Suicide', 'Suicide', 'Nephritis', 'Nephritis', 'Nephritis', 'Nephritis'),
levels=c('Alzheimers', 'Cancer', 'Cerebrovascular diseases', 'Diabetes', 'Flu and pneumonia', 'Heart disease', 'Lower respiratory', 'Nephritis', 'Suicide', 'Unintentional injuries')
)
Rate <- c(210.2, 242.7, 132.5, 154.9, 195.9, 219.3, 140.2, 150.8, 44.5, 62.8, 36.5, 46.9, 49.6, 71.3, 24.7, 37.2, 36.1, 42.2, 34.9, 42.2, 19.4, 21.8, 25.5, 30.6, 24.9, 29.5, 17.1, 21.8, 17.7, 20.8, 12.9, 16.3, 19.2, 26.3, 5.3, 6.2, 15.7, 18.3, 10.7, 13.9)
SE <- c(0.2, 0.6, 0.2, 0.4, 0.2, 0.5, 0.2, 0.4, 0.1, 0.3, 0.1, 0.2, 0.1, 0.3, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.1, 0.1, 0.2, 0, 0.1, 0.1, 0.2, 0, 0.1)
USMortality <- data.frame(Status=Status, Sex=Sex, Cause=Cause, Rate=Rate, SE=SE)
# Specify upper bound to exclude Heart disease and Cancer
x_limits <- c(0, 100)
# Draw the plot
dotplot(Cause ~ Rate | Sex + Status, data = USMortality, as.table = TRUE,
xlim = x_limits)
dotplot(Cause ~ Rate | Sex + Status, data = USMortality,
as.table = TRUE,
scales = list(x = list(relation = "free",
# Specify limits for each panel
limits = list(c(0, 50), c(0, 80),
c(0, 50), c(0, 80) ))))
dotplot(Cause ~ Rate | Sex + Status, data = USMortality,
as.table = TRUE,
# Change the number of tick marks
scales = list(x = list(tick.number = 10,
# Show `Rate` labels on both bottom and top
alternating = 3,
# Rotate `Rate` labels by 90 degrees
rot = 90),
# Rotate `Cause` labels by 45 degrees
y = list(rot = 45)))
# Define at as 2^3 up to 2^8
x_ticks_at <- 2 ** (3:8)
dotplot(Cause ~ Rate | Sex, data = USMortality,
groups = Status, auto.key = list(columns = 2),
scales = list(x = list(log = 2,
# A numeric vector with
# values 2^3, 2^4, ..., 2^8
at = x_ticks_at,
# A character vector,
# "8" for 2^3, "16" for 2^4, etc.
labels = x_ticks_at)))
# Create the dot plot
dotplot(Cause ~ Rate | Status, data = USMortality,
groups = Sex, auto.key = list(columns = 2),
scales = list(x = list(log = TRUE,
equispaced.log = FALSE)),
# Provide pch values for the two groups
pch = c(3, 1))
dotplot(Cause ~ Rate | Status, data = USMortality,
groups = Sex, auto.key = list(columns = 2),
par.settings = simpleTheme(pch = c(3, 1)),
scales = list(x = list(log = 2, equispaced.log = FALSE)))
# The WorldPhones matrix is already provided, with the first row removed so you only need consider consecutive years
data(WorldPhones)
WorldPhones <- WorldPhones[row.names(WorldPhones) != 1951, ]
WorldPhones## N.Amer Europe Asia S.Amer Oceania Africa Mid.Amer
## 1956 60423 29990 4708 2568 2366 1411 733
## 1957 64721 32510 5230 2695 2526 1546 773
## 1958 68484 35218 6662 2845 2691 1663 836
## 1959 71799 37598 6856 3000 2868 1769 911
## 1960 76036 40341 8220 3145 3054 1905 1008
## 1961 79831 43173 9053 3338 3224 2005 1076names(dimnames(WorldPhones)) <- c("Year", "Region")
# Transform matrix data to data frame
WorldPhonesDF <- as.data.frame(
# Intermediate step: convert to table
as.table(WorldPhones),
responseName = "Phones")
# Create the dot plot
dotplot(Year ~ Phones | Region,
data = WorldPhonesDF,
as.table = TRUE,
# Log-transform the x-axis
scales = list(x = list(log = TRUE,
equispaced.log = FALSE,
# Set x-axis relation to "sliced"
relation = "sliced")))
# Load latticeExtra package for ggplot2like()
library(latticeExtra)## Loading required package: RColorBrewer##
## Attaching package: 'latticeExtra'## The following object is masked _by_ '.GlobalEnv':
##
## USCancerRates## The following object is masked from 'package:ggplot2':
##
## layer# Transform matrix data to data frame
names(dimnames(WorldPhones)) <- c("Year", "Region")
WorldPhonesDF <-
as.data.frame(as.table(WorldPhones[-1, ]),
responseName = "Phones")
# Create the dot plot
dotplot(Year ~ Phones | Region,
data = WorldPhonesDF,
as.table = TRUE,
scales = list(x = list(log = TRUE,
equispaced.log = FALSE,
relation = "sliced")),
# Fill in suitable value of par.settings
par.settings = ggplot2like(),
# Fill in suitable value of lattice.options
lattice.options = ggplot2like.opts())
# Create factor variable
airquality$Month.Name <-
factor(airquality$Month, levels = 1:12,
labels = month.name[1:12])
# Create histogram of Ozone, conditioning on Month
histogram(~ Ozone | Month.Name,
data = airquality, as.table = TRUE,
# Set border to be transparent
border = "transparent",
# Set fill color to be mid-gray
col = "grey50")
# Create factor variable
airquality$Month.Name <-
factor(airquality$Month, levels = 1:12,
labels = month.name)
levels(airquality$Month.Name)## [1] "January" "February" "March" "April" "May"
## [6] "June" "July" "August" "September" "October"
## [11] "November" "December"# Drop empty levels
airquality$Month.Name <- droplevels(airquality$Month.Name)
levels(airquality$Month.Name)## [1] "May" "June" "July" "August" "September"# Obtain colors from RColorBrewer
library(RColorBrewer)
my.colors <- brewer.pal(n = 5, name = "Set1")
# Density plot of ozone concentration grouped by month
densityplot(~ Ozone, data = airquality, groups = Month.Name,
plot.points = FALSE,
auto.key = list(space = "right"),
# Fill in value of col
par.settings = simpleTheme(col = my.colors,
# Fill in value of lwd
lwd = 2))
Chapter 4 - Customizing plots using panel functions
Panel functions:
- Declarative approach (you provide specifications, system figures out requirements) is used by ggplot2
- Procedural approach (you provide step-by-step) is used by lattice and base R
- Custom displays in lattice cannot be created directly by faceting; instead, function build-up is needed
- panel.histdens <- function(x, …) { panel.histogram(x, …) ; panel.lines(density(x, na.rm=TRUE)) } # overlay a density with a density histogram
- The panel.histdens is then used inside the function, such as panel=panel.histdens inside a call to histogram()
- The base R functions like lines and points need to be replaced by their lattice equivalents like panel.lines and panel.points for the lattice code to work
- panel.histdens <- function(x, …) { panel.histogram(x, …) ; panel.lines(density(x, na.rm=TRUE)) } # overlay a density with a density histogram
- Rather than customizing a default display, sometimes you may want to replace it entirely
- An example of this is a violin plot, which is structured like a box and whisker plot, but instead of the boxes and whiskers, it uses kernel density estimates to summarize a distribution
- The resulting plot retains the compactness of a box and whisker plot, but also shows features like bimodality
- The built-in function panel.violin() in the lattice package implements the display of violin plots
- An example of this is a violin plot, which is structured like a box and whisker plot, but instead of the boxes and whiskers, it uses kernel density estimates to summarize a distribution
- When there are a large number of points in the data, there may be substantial overplotting in a standard scatter plot
- Another built-in panel function available in the lattice package that can serve as a replacement for panel.xyplot() in such cases is panel.smoothScatter()
- Instead of plotting the points directly, it uses a color gradient to show a 2-D kernel density estimate obtained from the data
- xyplot(rate.female ~ rate.male, data = USCancerRates, panel = panel.smoothScatter, scales = list(log = TRUE, equispaced.log = FALSE), main = “County-wise deaths due to cancer”)
Prepanel Functions to control limits:
- Controlling the x/y axis limits is enabled within lattice
- prepanel.histdens.1 <- function(x, …) { d <- density(x, na.rm = TRUE); list(ylim = c(0, max(d$y))) }
- histogram(~ rate.male + rate.female, USCancerRates, type = “density”, scales = list(x = list(log = 10)), xlab = “Rate”, panel = panel.histdens, prepanel = prepanel.histdens.1)
- prepanel.histdens.1 <- function(x, …) { d <- density(x, na.rm = TRUE); list(ylim = c(0, max(d$y))) }
Optional arguments of default panel functions:
- Some optional arguments are common to all high-level functions in lattice - xlab, ylab, main, layout, between, scales
- Some optional arguments are specific to a single high-level function
- nint, type - histogram()
- plot.points, ref - densityplot()
- grad, abline - xyplot()
- col, cwd, cex, pch
- nint, type - histogram()
- The high-level functions will handle the general arguments, while sweeping up all the others for passage to the panel functions
- For example, passing grid=TRUE in xyplot() passes the argument to panel.xyplot()
- For example, passing grid=TRUE in xyplot() passes the argument to panel.xyplot()
- The type argument for xyplot adds a number of arguments
- “p” - points
- “l” - lines
- “r” - regresion by way of panel.lmline()
- “smooth” - LOESS smooth by way of panel.loess()
- “a” - join average y values for each unique x value by way of panel.average()
- Multiple types can be specified as a vector
- “p” - points
- Note a few following features for an xyplot()
- grid = list(h=-1, v=0) # draws horizontal reference lines
- type = c(“p”, “a”) draws the points and a line connecting their averages
- jitter.x = TRUE will apply a jitter on the x-axis only
- grid = list(h=-1, v=0) # draws horizontal reference lines
- The default panel function for bwplot() has two additional arguments that you have not used before:
- pch = “|” replaces the black dot representing the median inside the box by a line segment dividing the box into two smaller rectangles
- notch = TRUE puts “notches” on the side of the boxes that indicate a confidence interval for median
- the overlapping of notches for two subgroups suggests that the true medians of the two subgroups are not significantly different
- pch = “|” replaces the black dot representing the median inside the box by a line segment dividing the box into two smaller rectangles
- For the last exercise in this chapter, your task is to recreate a grouped dot plot you have seen before, but replace the plotting characters by emoji images
- To do so, you will use the panel.xyimage() function in the latticeExtra package, which is similar to the panel.xyplot() function,
- except that plotting symbols are replaced by images whose locations (file names or URLs of JPEG or PNG image files) are specified as the pch argument
- To do so, you will use the panel.xyimage() function in the latticeExtra package, which is similar to the panel.xyplot() function,
Example code includes:
panel.xyrug <- function(x, y, ...)
{
# Reproduce standard scatter plot
panel.xyplot(x, y, ...)
# Identify observations with x-value missing
x.missing <- is.na(x)
# Identify observations with y-value missing
y.missing <- is.na(y)
# Draw rugs along axes
panel.rug(x = x[y.missing], y = y[x.missing])
}
airquality$Month.Name <-
factor(month.name[airquality$Month], levels = month.name)
xyplot(Ozone ~ Solar.R | Month.Name, data = airquality,
panel = panel.xyrug, as.table = TRUE)
# Create factor variable with month names
airquality$Month.Name <-
factor(month.name[airquality$Month], levels = month.name)
# Create box-and-whisker plot
bwplot(Month.Name ~ Ozone + Temp, airquality,
# Specify outer
outer=TRUE,
# Specify x-axis relation
scales = list(x = list(relation="free")),
# Specify layout
layout=c(2, 1),
# Specify x-axis label
xlab="Measured value")
# Create violin plot
bwplot(Month.Name ~ Ozone + Temp, airquality,
# Specify outer
outer = TRUE,
# Specify x-axis relation
scales = list(x = list(relation="free")),
# Specify layout
layout=c(2, 1),
# Specify x-axis label
xlab="Measured value",
# Replace default panel function
panel = panel.violin)
# Create panel function
panel.ss <- function(x, y, ...) {
# Call panel.smoothScatter()
panel.smoothScatter(x, y, ...)
# Call panel.loess()
panel.loess(x, y, col = "red")
# Call panel.abline()
panel.abline(0, 1)
}
# Create plot
xyplot(rate.female ~ rate.male, data = USCancerRates,
panel = panel.ss,
main = "County-wise deaths due to cancer")## (loaded the KernSmooth namespace)
# Define prepanel function
prepanel.histdens.2 <- function(x, ...) {
h <- prepanel.default.histogram(x, ...)
d <- density(x, na.rm = TRUE)
list(xlim = quantile(x, c(0.005, 0.995), na.rm = TRUE),
# Calculate upper y-limit
ylim = c(0, max(d$y, h$ylim[2])))
}
panel.histdens <- function(x, ...) {
panel.histogram(x, ...)
panel.lines(density(x, na.rm = TRUE))
}
# Create a histogram of rate.male and rate.female
histogram(~ rate.male + rate.female,
data = USCancerRates, outer = TRUE,
type = "density", nint = 50,
border = "transparent", col = "lightblue",
# The panel function: panel.histdens
panel = panel.histdens,
# The prepanel function: prepanel.histdens.2
prepanel = prepanel.histdens.2,
# Ensure that the x-axis is log-transformed
# and has relation "sliced"
scales = list(x = list(log = TRUE,
equispaced.log = FALSE,
relation = "sliced")),
xlab = "Rate (per 100,000)")
# Create the box and whisker plot
bwplot(division.ordered ~ rate.male,
data = USCancerRates,
# Indicate median by line instead of dot
pch = "|",
# Include notches for confidence interval
notch = TRUE,
# The x-axis should plot log-transformed values
scales = list(x = list(log=TRUE, equispaced.log=FALSE)),
xlab = "Death Rate in Males (per 100,000)")
# Load the 'latticeExtra' package
library(latticeExtra)
# Create summary dataset
USCancerRates.state <-
with(USCancerRates, {
rmale <- tapply(rate.male, state, median, na.rm = TRUE)
rfemale <- tapply(rate.female, state, median, na.rm = TRUE)
data.frame(Rate = c(rmale, rfemale),
State = rep(names(rmale), 2),
Gender = rep(c("Male", "Female"),
each = length(rmale)))
})
# Reorder levels
library(dplyr)
USCancerRates.state <-
mutate(USCancerRates.state, State = reorder(State, Rate))
head(USCancerRates.state)## Rate State Gender
## 1 286.00 Alabama Male
## 2 237.95 Alaska Male
## 3 209.30 Arizona Male
## 4 284.10 Arkansas Male
## 5 221.30 California Male
## 6 204.40 Colorado Male# URLs for emojis
emoji.man <- "https://twemoji.maxcdn.com/72x72/1f468.png"
emoji.woman <- "https://twemoji.maxcdn.com/72x72/1f469.png"
# Create dotplot
# dotplot(State ~ Rate, data = USCancerRates.state,
# Specify grouping variable
# groups = Gender,
# Specify panel function
# panel = panel.xyimage,
# Specify emoji URLs
# pch = c(emoji.woman, emoji.man),
# Make symbols smaller
# cex = 0.75)Chapter 5 - Extensions and the lattice ecosystem
New methods - lattice is used by many packages because it is highly extensible:
- High-level lattice functions are “generic functions”, and the first argument need not be a formula
- For example, dotplot() can be applied directly to a table
- For example, dotPlot(worldPhones[-1, ], scales=list(x=list(log=2)), groups=FALSE, layout=c(1, NA), strip=FALSE, strip.left=TRUE)
- For example, dotPlot(worldPhones[-1, ], scales=list(x=list(log=2)), groups=FALSE, layout=c(1, NA), strip=FALSE, strip.left=TRUE)
- The xyplot() function has a suitable method for time series objects
- The function to create the time-series plot is simply xyplot()
- Instead of a formula and a data frame, the only mandatory argument is a time series object, which must be the first argument
- The default value of type is “l”, so that data points are joined by lines
- The argument superpose, which can take values TRUE or FALSE, is used to control whether multiple time series are plotted within the same panel or in separate panels, respectively
- The default is to plot them separately
- The argument cut, which should be a list of the form list(number = , overlap = ), is used to produce so-called “cut-and-stack” plots, by splitting the time axis into multiple overlapping periods which are then used to condition
- This makes it easier to see parts of a long series
- The function to create the time-series plot is simply xyplot()
- One innovative display design for time series data, known as horizon graphs, is implemented in the panel.horizonplot() function in the latticeExtra package
- Horizon plots allow you to visualize many time series in a small amount of space
- The main motivation for this design is to reduce the vertical space occupied by a single time series, without the loss of resolution that would result from simply flattening the usual line graph display
- This is achieved in two ways. First, negative values are mirrored to lie above the x-axis, but distinguished from positive values by shading using different colors
- Second, values are divided into bands with progressively higher saturation, and the bands are collapsed to wrap them around lower bands
- Horizon plots allow you to visualize many time series in a small amount of space
New high-level functions can be created:
- Completely new high-level functions are built when the panel options are insufficient
- The horizonplot() for above is one example
- The chloropleth (colored map) is another - see mapplot() in the latticeExtra() package
- The horizonplot() for above is one example
- Since the earth is three dimensional but the plot is two dimensional, a projection is required to reduce the number of dimensions
- The list of available projections is given in the Details section of the mapproject() help page
- The list of available projections is given in the Details section of the mapproject() help page
- Map plots are drawn in two stages. First, a map object is created using the map() function from the maps package with plot = FALSE
- the_map <- map(“a_map_dataset”, plot = FALSE, projection = “some_projection”)
- the_map <- map(“a_map_dataset”, plot = FALSE, projection = “some_projection”)
- Second, mapplot() is called with a formula, a data frame, and a map
- mapplot(region ~ value, data, map = the_map)
- mapplot(region ~ value, data, map = the_map)
- It is common to have statistical estimates in the form of confidence intervals in addition to point estimates
- Such data can be displayed using segment plots via the segplot() function in the latticeExtra package
segplot( - categories ~ lower_limit + upper_limit, data = some_data, centers = point_estimates)
- Notice that the categories are displayed on the y-axis, and the confidence intervals are displayed on the x-axis
- The point estimates, usually a mean or median value for that category, are specified using the centers argument, not the formula
- An optional argument, draw.bands, let’s you choose between confidence bands and confidence intervals
- This argument is passed to the default panel function panel.segplot()
- Such data can be displayed using segment plots via the segplot() function in the latticeExtra package
- One common approach is to plot some form of bivariate density estimate instead of the raw data, as is done with histograms and kernel density plots for univariate data
- Hexagonal binning and plotting is implemented in the R package hexbin, which also includes the high-level function hexbinplot() for creating conditional hexbin plots using the lattice framework
- The formula and data argument in a hexbinplot() call is interpreted in the same way as xyplot()
- The type argument can be set to “r” to add a regression line
- The trans argument can be a function that is applied to the observed counts before creating bands for different colors
- By default, the range of counts is divided up evenly into bands, but taking the square root of the counts, for example, emphasizes differences in the lower range of counts more
- The inv argument gives the inverse function of trans, so that transformed counts can be converted back before being shown in the legend
- Hexagonal binning and plotting is implemented in the R package hexbin, which also includes the high-level function hexbinplot() for creating conditional hexbin plots using the lattice framework
Manipulation (extension) of trellis objects:
- latticeExtra::useOuterStrip(latticeObject) will make the strips show only on the top and the left
- The directlabels package tackles an interesting problem: instead of having a separate legend associating graphical parameters and levels of a grouping variable, it tries to indicate the grouping by placing text labels within the panel
- This is generally a tricky thing to do automatically. directlabels relies on heuristics, and also allows the user to provide their own heuristics. It works with both lattice and ggplot2 plots
- This is generally a tricky thing to do automatically. directlabels relies on heuristics, and also allows the user to provide their own heuristics. It works with both lattice and ggplot2 plots
- Once a lattice plot object is created, it can be modified using the update() method
- Among other things, a new panel function can be provided as the panel argument, to change or enhance the panel display
- Specifying the display in the form of a function can be cumbersome, especially for minor changes
- An alternative approach, implemented in the latticeExtra package, is to add so-called layers to the existing display. This is modeled on the approach used by the ggplot2 package
- Among other things, a new panel function can be provided as the panel argument, to change or enhance the panel display
- There are two kinds of layers
- Layers that go below the default display (i.e., are drawn before it) are created by the layer_() function
- Those that go above are created using layer()
- There are also corresponding versions glayer_() and glayer() for grouped displays
- A layer is created by putting a function call, as it would appear inside a panel function, inside a call to layer_() or layer()
- Layers that go below the default display (i.e., are drawn before it) are created by the layer_() function
- Suppose you want to create a layer with a call to panel.grid that goes under the display, and a call to panel.lmline() that goes above, and then add it to an existing lattice plot p
- under_layer <- layer_(panel.grid())
- over_layer <- layer(panel.lmline(x, y))
- p + under_layer + over_layer
- Layers are added to a plot using the + operator
Example code includes:
# Use 'EuStockMarkets' time series data
data(EuStockMarkets)
str(EuStockMarkets)## Time-Series [1:1860, 1:4] from 1991 to 1999: 1629 1614 1607 1621 1618 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:4] "DAX" "SMI" "CAC" "FTSE"# Create time series plot
xyplot(EuStockMarkets,
# Plot all series together
superpose = TRUE,
# Split up the time axis into parts
cut = list(number = 3, overlap = 0.25))
# Create time series plot
xyplot(EuStockMarkets,
# Specify panel function
panel=panel.horizonplot,
# Specify prepanel function
prepanel=prepanel.horizonplot)
# Load required packages
library(maps)
# Create map object for US counties
county.map <- map("county", plot = FALSE, fill = TRUE,
# Specify projection
projection = "sinusoidal")
# Create choropleth map
row.names(USCancerRates) <- rn_USCR
mapplot(row.names(USCancerRates) ~ log10(rate.male) + log10(rate.female),
data = USCancerRates,
xlab = "", scales = list(draw = FALSE),
# Specify map
map = county.map)## Warning in (function (x, y, map, breaks, colramp, exact = FALSE,
## lwd = 0.5, : 64 unmatched regions: alaska,nome, alaska,wade hampton,
## alaska,haines, alaska,....## Warning in (function (x, y, map, breaks, colramp, exact = FALSE,
## lwd = 0.5, : 64 unmatched regions: alaska,nome, alaska,wade hampton,
## alaska,haines, alaska,....
# Create subset for Louisiana
LACancerRates1 <- filter(USCancerRates, state == "Louisiana")
str(LACancerRates1)## 'data.frame': 64 obs. of 11 variables:
## $ rate.male : num 369 361 349 338 338 ...
## $ LCL95.male : num 316 289 302 308 303 ...
## $ UCL95.male : num 428 446 402 372 376 ...
## $ rate.female : num 162 193 215 194 192 ...
## $ LCL95.female : num 134 150 184 176 170 ...
## $ UCL95.female : num 196 246 250 215 218 ...
## $ state : Factor w/ 49 levels "Alabama","Alaska",..: 17 17 17 17 17 17 17 17 17 17 ...
## $ county :Class 'AsIs' chr [1:64] "Richland Parish" "Madison Parish" "De Soto Parish" "St. Bernard Parish" ...
## $ state.ordered : Factor w/ 49 levels "Utah","New Mexico",..: 46 46 46 46 46 46 46 46 46 46 ...
## ..- attr(*, "scores")= num [1:49(1d)] 166 166 145 169 159 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "Alabama" "Alaska" "Arizona" "Arkansas" ...
## $ division : Factor w/ 9 levels "New England",..: 5 5 5 5 5 5 5 5 5 5 ...
## $ division.ordered: Factor w/ 9 levels "Mountain","West North Central",..: 7 7 7 7 7 7 7 7 7 7 ...
## ..- attr(*, "scores")= num [1:9(1d)] 417 416 446 466 433 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "New England" "Middle Atlantic" "South Atlantic" "East South Central" ...# Reorder levels of county
LACancerRates2 <-
mutate(LACancerRates1,
county = reorder(county, rate.male))
# Draw confidence intervals
segplot(county ~ LCL95.male + UCL95.male,
data = LACancerRates2,
# Add point estimates
centers = rate.male,
# Draw segments rather than bands
draw.bands = FALSE)
# The 'USCancerRates' dataset
str(USCancerRates)## 'data.frame': 3041 obs. of 11 variables:
## $ rate.male : num 364 346 341 336 330 ...
## $ LCL95.male : num 311 274 304 289 293 ...
## $ UCL95.male : num 423 431 381 389 371 ...
## $ rate.female : num 151 140 182 185 172 ...
## $ LCL95.female : num 124 103 161 157 151 ...
## $ UCL95.female : num 184 190 206 218 195 ...
## $ state : Factor w/ 49 levels "Alabama","Alaska",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ county :Class 'AsIs' chr [1:3041] "Pickens County" "Bullock County" "Russell County" "Barbour County" ...
## $ state.ordered : Factor w/ 49 levels "Utah","New Mexico",..: 25 25 25 25 25 25 25 25 25 25 ...
## ..- attr(*, "scores")= num [1:49(1d)] 166 166 145 169 159 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "Alabama" "Alaska" "Arizona" "Arkansas" ...
## $ division : Factor w/ 9 levels "New England",..: 4 4 4 4 4 4 4 4 4 4 ...
## $ division.ordered: Factor w/ 9 levels "Mountain","West North Central",..: 9 9 9 9 9 9 9 9 9 9 ...
## ..- attr(*, "scores")= num [1:9(1d)] 417 416 446 466 433 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "New England" "Middle Atlantic" "South Atlantic" "East South Central" ...# Load the 'hexbin' package
library(hexbin)
# Create hexbin plot
hexbinplot(rate.female ~ rate.male,
data = USCancerRates,
# Add a regression line
type = "r",
# function to transform counts
trans = sqrt,
# function to invert transformed counts
inv = function(x) x^2
)
# Load the 'directlabels' package
library(directlabels)
# Use the 'airquality' dataset
str(airquality)## 'data.frame': 153 obs. of 7 variables:
## $ Ozone : int 41 36 12 18 NA 28 23 19 8 NA ...
## $ Solar.R : int 190 118 149 313 NA NA 299 99 19 194 ...
## $ Wind : num 7.4 8 12.6 11.5 14.3 14.9 8.6 13.8 20.1 8.6 ...
## $ Temp : int 67 72 74 62 56 66 65 59 61 69 ...
## $ Month : int 5 5 5 5 5 5 5 5 5 5 ...
## $ Day : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Month.Name: Factor w/ 12 levels "January","February",..: 5 5 5 5 5 5 5 5 5 5 ...# Create factor variable
airquality$Month.Name <-
factor(month.name[airquality$Month], levels = month.name)
# Create density plot object
tplot2 <-
densityplot(~ Ozone + Temp, data = airquality,
# Variables should go in different panels
outer = TRUE,
# Specify grouping variable
groups = Month.Name,
# Suppress display of data points
plot.points = FALSE,
# Add reference line
ref = TRUE,
# Specify layout
layout = c(2, 1),
# Omit strip labels
strip = FALSE,
# Provide column-specific x-axis labels
xlab = c("Ozone (ppb)", "Temperature (F)"),
# Let panels have independent scales
scales = list(relation="free"))
# Produce plot with direct labels
direct.label(tplot2)
# 'USCancerRates' is pre-loaded
str(USCancerRates)## 'data.frame': 3041 obs. of 11 variables:
## $ rate.male : num 364 346 341 336 330 ...
## $ LCL95.male : num 311 274 304 289 293 ...
## $ UCL95.male : num 423 431 381 389 371 ...
## $ rate.female : num 151 140 182 185 172 ...
## $ LCL95.female : num 124 103 161 157 151 ...
## $ UCL95.female : num 184 190 206 218 195 ...
## $ state : Factor w/ 49 levels "Alabama","Alaska",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ county :Class 'AsIs' chr [1:3041] "Pickens County" "Bullock County" "Russell County" "Barbour County" ...
## $ state.ordered : Factor w/ 49 levels "Utah","New Mexico",..: 25 25 25 25 25 25 25 25 25 25 ...
## ..- attr(*, "scores")= num [1:49(1d)] 166 166 145 169 159 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "Alabama" "Alaska" "Arizona" "Arkansas" ...
## $ division : Factor w/ 9 levels "New England",..: 4 4 4 4 4 4 4 4 4 4 ...
## $ division.ordered: Factor w/ 9 levels "Mountain","West North Central",..: 9 9 9 9 9 9 9 9 9 9 ...
## ..- attr(*, "scores")= num [1:9(1d)] 417 416 446 466 433 ...
## .. ..- attr(*, "dimnames")=List of 1
## .. .. ..$ : chr "New England" "Middle Atlantic" "South Atlantic" "East South Central" ...# Create scatter plot
p <- xyplot(rate.female ~ rate.male, data = USCancerRates,
# Change plotting character
pch = 16,
# Make points semi-transparent
alpha = 0.25)
# Create layer with reference grid
l0 <- layer_(panel.grid())
# Create layer with reference line
l1 <- layer(panel.abline(0, 1))
# Create layer with regression fit
l2 <- layer(panel.smoother(x, y, method="lm"))
# Combine and plot
p + l0 + l1 + l2
Visualizing Time Series Data in R
Chapter 1 - R Time Series Visualization Tools
Refresher on xts and the plot() function:
- With a time series plot, each element is associated to a specific time
- The xts objects is typically the storage mechanism for times series data in R
- Time Index (Date, POSIXct, or the like) + Matrix
- Time Index (Date, POSIXct, or the like) + Matrix
- The plot() call can be used on xts objects and will call plot.xts() to achieve this purpose
- Many of the calls are similar to a normal plot() - for example, can overwrite using lines()
Other useful visualizing functions:
- Can use lines() to add a line to an existing time series plot
- Can use axis(side=, at=) # 1 bottom, 2 left, 3 top, 4 right ; can use at=pretty(existingPlotData)
- Can add legends using legend(x=
, legend=, col=, lty=)
- Can add lines to a plot using abline(v=, h=)
- The PerformanceAnalytics package allows for better highlighting portions of the plot
- To highlight a specific period in a time series, you can display it in the plot in a different background color
- The chart.TimeSeries() function in the PerformanceAnalytics package offers a very easy and flexible way of doing this
- chart.TimeSeries(R, period.areas, period.color)
- R is an xts, time series, or zoo object of asset returns
- period.areas are shaded areas specified by a start and end date in a vector of xts date ranges like c(“1926-10/1927-11”)
- period.color draws the shaded region in whichever color is specified
- The chart.TimeSeries() function in the PerformanceAnalytics package offers a very easy and flexible way of doing this
Example code includes:
library(xts)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric##
## Attaching package: 'xts'## The following objects are masked from 'package:dplyr':
##
## first, last# data is a 504x4 xts object of Yahoo, Microsoft, Citigroup, and Dow
tmpData <- readr::read_delim("./RInputFiles/dataset_1_1.csv", delim=" ")## Parsed with column specification:
## cols(
## Index = col_date(format = ""),
## yahoo = col_double(),
## microsoft = col_double(),
## citigroup = col_double(),
## dow_chemical = col_double()
## )data <- xts::xts(tmpData[, -1], order.by=as.POSIXct(tmpData$Index))
# Display the first few lines of the data
head(data)## yahoo microsoft citigroup dow_chemical
## 2015-01-01 18:00:00 50.17 44.30501 53.45259 42.48209
## 2015-01-04 18:00:00 49.13 43.89759 51.76803 41.16821
## 2015-01-05 18:00:00 49.21 43.25329 49.94556 40.50662
## 2015-01-06 18:00:00 48.59 43.80284 50.40857 40.44139
## 2015-01-07 18:00:00 50.23 45.09144 51.16711 41.44776
## 2015-01-08 18:00:00 49.72 44.71244 50.02437 41.38253# Display the column names of the data
colnames(data)## [1] "yahoo" "microsoft" "citigroup" "dow_chemical"# Plot yahoo data and add title
plot(data[, "yahoo"], main="yahoo")
# Replot yahoo data with labels for X and Y axes
plot(data[, "yahoo"], main="yahoo", xlab="date", ylab="price")
# Note that type="h" is for bars
# Plot the second time series and change title
plot(data[, 2], main="microsoft")
# Replot with same title, add subtitle, use bars
plot(data[, 2], main="microsoft", sub="Daily closing price since 2015", type="h")
# Change line color to red
lines(data[, 2], col="red")
# Plot two charts on same graphical window
par(mfrow = c(2, 1))
plot(data[, 1], main="yahoo")
plot(data[, 2], main="microsoft")
# Replot with reduced margin and character sizes
par(mfrow = c(2, 1), mex=0.6, cex=0.8)
plot(data[, 1], main="yahoo")
plot(data[, 2], main="microsoft")
par(mfrow = c(1, 1), mex=1, cex=1)
# Plot the "microsoft" series
plot(data[, "microsoft"], main="Stock prices since 2015")
# Add the "dow_chemical" series in red
lines(data[, "dow_chemical"], col="red")
# Add a Y axis on the right side of the chart
axis(side=4, at=pretty(data[, "dow_chemical"]))
# Add a legend in the bottom right corner
legend("bottomright", legend=c("microsoft", "dow_chemical"), col=c("black", "red"), lty=c(1, 1))
# Plot the "citigroup" time series
plot(data[, "citigroup"], main="Citigroup")
# Create vert_line to identify January 4th, 2016 in citigroup
vert_line <- which(index(data[, "citigroup"]) == as.POSIXct("2016-01-04"))
# Add a red vertical line using vert_line
abline(v = .index(data[, "citigroup"])[vert_line], col = "red")
# Create hori_line to identify average price of citigroup
hori_line <- mean(data[, "citigroup"])
# Add a blue horizontal line using hori_line
abline(h = hori_line, col = "blue")
# Create period to hold the 3 months of 2015
period <- c("2015-01/2015-03")
# Highlight the first three months of 2015
PerformanceAnalytics::chart.TimeSeries(data[, "citigroup"], period.areas=period)
# Highlight the first three months of 2015 in light grey
PerformanceAnalytics::chart.TimeSeries(data[, "citigroup"], period.areas=period, period.color="lightgrey")
# Plot the microsoft series
plot(data[, "microsoft"], main="Dividend date and amount")
# Add the citigroup series
lines(data[, "citigroup"], col="orange", lwd=2)
# Add a new y axis for the citigroup series
axis(side=4, at=pretty(data[, "citigroup"]), col="orange")
micro_div_date <- "15 Nov. 2016"
citi_div_date <- "13 Nov. 2016"
micro_div_value <- "$0.39"
citi_div_value <- "$0.16"
# Same plot as the previous exercise
plot(data$microsoft, main = "Dividend date and amount")
lines(data$citigroup, col = "orange", lwd = 2)
axis(side = 4, at = pretty(data$citigroup), col = "orange")
# Create the two legend strings
micro <- paste0("Microsoft div. of ", micro_div_value," on ", micro_div_date)
citi <- paste0("Citigroup div. of ", citi_div_value," on ", citi_div_date)
# Create the legend in the bottom right corner
legend(x = "bottomright", legend = c(micro, citi), col = c("black", "orange"), lty = c(1, 1))
data_1_1_old <- dataChapter 2 - Univariate Time Series
Univariate time series analysis - deals with only a single variable:
- Location, Dispersion, Distribution - frequently presented by way of histograms
- Time series typically need to be transformed prior to these calculations, since their data is in the wrong format otherwise
- For example, it is often more helpful to get the distribution of price change (and/or percentage return) rather than just the stock price
- For example, it is often more helpful to get the distribution of price change (and/or percentage return) rather than just the stock price
- In finance, price series are often transformed to differenced data, making it a return series
- In R, the ROC() (which stands for “Rate of Change”) function from the TTR package does this automatically to a price or volume series x
Other visualization tools:
- Can create histograms of stock returns
- Can use boxplot() to see the box-and-whisker of the stock returns
- The argument horizontal=TRUE will display the block horizontally
- The argument horizontal=TRUE will display the block horizontally
- Can run acf() to see the autocorrelation of the returns
- Can run qqnorm() and qqline() to see whether the data are normally distributed
Combining everything so far:
- The histogram helps with understanding both central tendencies and outliers
- The box and whiskers plot helps in a similar manner - also helps to show investment riskiness
- The box and whiskers plot helps in a similar manner - also helps to show investment riskiness
- The autocorrelation plot helps with understanding the linkages between today and days in the future
- The QQ plot helps to assess whether methods/tests that rely on normality can be safely used on the dataset
Example code includes:
tmpData <- readr::read_delim("./RInputFiles/dataset_2_1.csv", delim=" ")## Parsed with column specification:
## cols(
## Index = col_date(format = ""),
## Apple = col_double()
## )names(tmpData) <- c("Index", "apple")
data <- xts::xts(tmpData[, -1], order.by=as.Date(tmpData$Index))
# indexClass(data) <- c("POSIXt", "POSIXlt")
# Plot Apple's stock price
plot(data[, "apple"], main="Apple stock price")
# Create a time series called rtn
rtn <- TTR::ROC(data[, "apple"])
# Plot Apple daily price and daily returns
par(mfrow=c(1, 2))
plot(data[, "apple"], main="Apple stock price")
plot(rtn)
par(mfrow=c(1, 1))
dim(rtn)## [1] 522 1rtn <- rtn[complete.cases(rtn), ]
dim(rtn)## [1] 521 1# Create a histogram of Apple stock returns
hist(rtn, main="Apple stock return distribution", probability=TRUE)
# Add a density line
lines(density(rtn[complete.cases(rtn), ]))
# Redraw a thicker, red density line
lines(density(rtn[complete.cases(rtn), ]), col="red", lwd=2)
rtnRaw <- as.double(rtn$apple)
# Draw box and whisker plot for the Apple returns
boxplot(rtnRaw)
# Draw a box and whisker plot of a normal distribution
boxplot(rnorm(1000))
# Redraw both plots on the same graphical window
par(mfrow=c(2, 1))
boxplot(rtnRaw, horizontal=TRUE)
boxplot(rnorm(1000), horizontal=TRUE)
par(mfrow=c(1, 1))
# Draw autocorrelation plot
acf(rtn, main="Apple return autocorrelation")
# Redraw with a maximum lag of 10
acf(rtn, main="Apple return autocorrelation", lag.max=10)
# Create q-q plot
qqnorm(rtn, main="Apple return QQ-plot")
# Add a red line showing normality
qqline(rtn, col="red")
par(mfrow=c(2, 2))
hist(rtn, probability=TRUE)
lines(density(rtn), col="red")
boxplot(rtnRaw)
acf(rtn)
qqnorm(rtn)
qqline(rtn, col="red")
par(mfrow=c(1, 1))Chapter 3 - Multivariate Time Series
Dealing with higher dimensions - visualization challenges with larger numbers of series:
- Might want to compare stock prices vs interest rate changes
- Cannot easily visualize even 10 time series, let alone 100 time series
- One solution is to plot both time series as barcharts. There are two types:
- Grouped barchart: for a single period, there are as many bars as time series
- Stacked bar chart: for each period, there is a single bar, and each time series is represented by a portion of the bar proportional to the value of the time series at this date (i.e. the total at each period adds up to 100%)
- Grouped barchart: for a single period, there are as many bars as time series
Multivariate time series:
- To create a stacked chart, use barchart(myFrame, col=c(), main=) # can specify the desired colors in the barchart or use the defaults
- Can create the correlation matrix using cor(myMatrix, digit=)
- Several types of correlations exist but the most used ones are:
- Pearson correlation: measures the linear relationship between 2 variables
- Spearman rank correlation: measures the statistical dependency between the ranking of 2 variables (not necessarily linear)
- Several types of correlations exist but the most used ones are:
- Can create the pair chart using pairs(myFrame, lower.panel=NULL, main=) # the lower.panel=NULL shows only the diagonal and the upper-right of the pairs plot
- Can create a correlation plot using corrplot(myMatrix, method=“number”, type=“upper”) # type=“upper” shows only the upper-right of the diagonal
Higher dimension time series:
- Can display a correlation matrix as a heat map
- corrplot(myMatrix, method=“color”, type=“upper”)
Example code includes:
# You are provided with a dataset (portfolio) containing the weigths of stocks A (stocka) and B (stockb) in your portfolio for each month in 2016
stockA <- c(0.1, 0.4, 0.5, 0.5, 0.2, 0.3, 0.7, 0.8, 0.7, 0.2, 0.1, 0.2)
stockB <- c(0.9, 0.6, 0.5, 0.5, 0.8, 0.7, 0.3, 0.2, 0.3, 0.8, 0.9, 0.8)
pDates <- as.Date(c('2016-01-01', '2016-02-01', '2016-03-01', '2016-04-01', '2016-05-01', '2016-06-01', '2016-07-01', '2016-08-01', '2016-09-01', '2016-10-01', '2016-11-01', '2016-12-01'))
portfolio <- xts(data.frame(stocka=stockA, stockb=stockB), order.by=pDates)
# Plot stacked barplot
barplot(portfolio)
# Plot grouped barplot
barplot(portfolio, beside=TRUE)
tmpData <- readr::read_delim("./RInputFiles/data_3_2.csv", delim=",")## Parsed with column specification:
## cols(
## Index = col_date(format = ""),
## sp500 = col_double(),
## citigroup = col_double(),
## microsoft = col_double(),
## apple = col_double(),
## dowchemical = col_double(),
## yahoo = col_double()
## )# names(tmpData) <- c("Index", "apple")
my_data <- xts::xts(tmpData[, -1], order.by=as.Date(tmpData$Index))
citi <- as.numeric(my_data$citigroup)
sp500 <- as.numeric(my_data$sp500)
# Draw the scatterplot
plot(y=citi, x=sp500)
# Draw a regression line
abline(reg=lm(citi ~ sp500), col="red", lwd=2)
# my_data containing the returns for 5 stocks: ExxonMobile, Citigroup, Microsoft, Dow Chemical and Yahoo
# Create correlation matrix using Pearson method
cor(my_data)## sp500 citigroup microsoft apple dowchemical yahoo
## sp500 1.0000000 0.5097953 0.3743215 0.3576966 0.5217243 0.2900962
## citigroup 0.5097953 1.0000000 0.4841408 0.4291841 0.5085190 0.4029490
## microsoft 0.3743215 0.4841408 1.0000000 0.5133469 0.3954523 0.4329388
## apple 0.3576966 0.4291841 0.5133469 1.0000000 0.3627755 0.3413626
## dowchemical 0.5217243 0.5085190 0.3954523 0.3627755 1.0000000 0.2938749
## yahoo 0.2900962 0.4029490 0.4329388 0.3413626 0.2938749 1.0000000# Create correlation matrix using Spearman method
cor(my_data, method="spearman")## sp500 citigroup microsoft apple dowchemical yahoo
## sp500 1.0000000 0.5192579 0.4244237 0.3518853 0.5316235 0.3262037
## citigroup 0.5192579 1.0000000 0.4976477 0.4374850 0.5607511 0.3780730
## microsoft 0.4244237 0.4976477 1.0000000 0.5128477 0.4684114 0.4448179
## apple 0.3518853 0.4374850 0.5128477 1.0000000 0.3681791 0.3680715
## dowchemical 0.5316235 0.5607511 0.4684114 0.3681791 1.0000000 0.3464743
## yahoo 0.3262037 0.3780730 0.4448179 0.3680715 0.3464743 1.0000000# Create scatterplot matrix
pairs(as.data.frame(my_data))
# Create upper panel scatterplot matrix
pairs(as.data.frame(my_data), lower.panel=NULL)
cor_mat <- cor(my_data)
# In this exercise, you will use the provided correlation matrix cor_mat
# Create correlation matrix
corrplot::corrplot(cor_mat)
# Create correlation matrix with numbers
corrplot::corrplot(cor_mat, method="number")
# Create correlation matrix with colors
corrplot::corrplot(cor_mat, method="color")
# Create upper triangle correlation matrix
corrplot::corrplot(cor_mat, method="number", type="upper")
# Draw heatmap of cor_mat
corrplot::corrplot(cor_mat, method="color")
# Draw upper heatmap
corrplot::corrplot(cor_mat, method="color", type="upper")
# Draw the upper heatmap with hclust
corrplot::corrplot(cor_mat, method="color", type="upper", order="hclust")
Chapter 4 - Case Study: Stock Picking for Portfolios
Case study presentation:
- Suppose you have a portfolio of Apple, Microsoft, and Yahoo
- Suppose also that you can add just a single extra stock with some spare cash
- Examine the correlations of new stocks to the existing portfolio
- Starting point assumption is capital protection - low correlation to the existing portfolio
- Starting point assumption is capital protection - low correlation to the existing portfolio
- The PerformanceAnalytics package has some helpful tools for this analysis
New stocks:
- Goal is to choose the best new stock for the portfolio
- The PerformanceAnalytics package provides additional tools to get a finer view of your portfolio
- In particular, the charts.PerformanceSummary() function provides a quick and easy way to display the portfolio value, returns and periods of poor performance, also known as drawdowns
Course conclusion:
- xts, plot()
- Univariate
- Multivariate
- Case study
Example code includes:
# In this exercise, you are provided with a dataset data containing the value and the return of the portfolio over time, in value and return, respectively.
tmpData <- readr::read_delim("./RInputFiles/data_4_1.csv", delim=",")## Parsed with column specification:
## cols(
## Index = col_date(format = ""),
## value = col_double(),
## return = col_double()
## )# names(tmpData) <- c("Index", "apple")
data <- xts::xts(tmpData[, -1], order.by=as.Date(tmpData$Index))
# indexClass(data) <- c("POSIXt", "POSIXlt")
# Plot the portfolio value
plot(data$value, main="Portfolio Value")
# Plot the portfolio return
plot(data$return, main="Portfolio Return")
# Plot a histogram of portfolio return
hist(data$return, probability=TRUE)
# Add a density line
lines(density(data$return), col="red", lwd=2)
tmpPortfolioData <- data
# The new dataset data containing four new stocks is available in your workspace: Goldman Sachs (GS), Coca-Cola (KO), Walt Disney (DIS), Caterpillar (CAT)
tmpData <- readr::read_delim("./RInputFiles/data_4_3.csv", delim=",")## Parsed with column specification:
## cols(
## Index = col_date(format = ""),
## GS = col_double(),
## KO = col_double(),
## DIS = col_double(),
## CAT = col_double()
## )# names(tmpData) <- c("Index", "apple")
data <- xts::xts(tmpData[, -1], order.by=as.Date(tmpData$Index))
# indexClass(data) <- c("POSIXt", "POSIXlt")
# Plot the four stocks on the same graphical window
par(mfrow=c(2, 2), mex=0.8, cex=0.8)
plot(data[, 1])
plot(data[, 2])
plot(data[, 3])
plot(data[, 4])
par(mfrow=c(1, 1), mex=1, cex=1)
# In this exercise, you are provided with four individual series containing the return of the same four stocks:
# gs, ko, dis, cat
# Solution makes absolutely no sense
portfolio <- as.numeric(tmpPortfolioData$return)
gs <- as.numeric(TTR::ROC(data[, "GS"]))[-1]
ko <- as.numeric(TTR::ROC(data[, "KO"]))[-1]
dis <- as.numeric(TTR::ROC(data[, "DIS"]))[-1]
cat <- as.numeric(TTR::ROC(data[, "CAT"]))[-1]
# Draw the scatterplot of gs against the portfolio
plot(y=portfolio, x=gs)
# Add a regression line in red
abline(reg=lm(gs ~ portfolio), col="red", lwd=2)
# Plot scatterplots and regression lines to a 2x2 window
par(mfrow=c(2, 2))
plot(y=portfolio, x=gs)
abline(reg=lm(gs ~ portfolio), col="red", lwd=2)
plot(y=portfolio, x=ko)
abline(reg=lm(ko ~ portfolio), col="red", lwd=2)
plot(y=portfolio, x=dis)
abline(reg=lm(dis ~ portfolio), col="red", lwd=2)
plot(y=portfolio, x=cat)
abline(reg=lm(cat ~ portfolio), col="red", lwd=2)
par(mfrow=c(1, 1))
# In this exercise, you are given a dataset old.vs.new.portfolio with the following self-explanatory columns: old.portfolio.value, new.portfolio.value, old.portfolio.rtn, new.portfolio.rtn
tmpData <- readr::read_delim("./RInputFiles/old.vs.new.portfolio.csv", delim=",")## Parsed with column specification:
## cols(
## Index = col_date(format = ""),
## old.portfolio.value = col_double(),
## new.portfolio.value = col_double(),
## old.portfolio.rtn = col_double(),
## new.portfolio.rtn = col_double()
## )# names(tmpData) <- c("Index", "apple")
old.vs.new.portfolio <- xts::xts(tmpData[, -1], order.by=as.Date(tmpData$Index))
# indexClass(data) <- c("POSIXt", "POSIXlt")
# Plot new and old portfolio values on same chart
plot(old.vs.new.portfolio$old.portfolio.value)
lines(old.vs.new.portfolio$new.portfolio.value, col = "red")
# Plot density of the new and old portfolio returns on same chart
plot(density(old.vs.new.portfolio$old.portfolio.rtn))
lines(density(old.vs.new.portfolio$new.portfolio.rtn), col ="red")
# Draw value, return, drawdowns of old portfolio
PerformanceAnalytics::charts.PerformanceSummary(old.vs.new.portfolio[, "old.portfolio.rtn"])
# Draw value, return, drawdowns of new portfolio
PerformanceAnalytics::charts.PerformanceSummary(old.vs.new.portfolio[, "new.portfolio.rtn"])
# Draw both portfolios on same chart
# Draw value, return, drawdowns of new portfolio
PerformanceAnalytics::charts.PerformanceSummary(old.vs.new.portfolio[, c("old.portfolio.rtn", "new.portfolio.rtn")])
Communicating with the Tidyverse
Chapter 1 - Custom ggplot2 themes
Introduction to the data - finding stories in datasets:
- Communication is the final step in the tidyverse workflow
- This course will create a production-level plot from Swiss public radio regarding hours worked in Europe, using ggplot2
- Will also create a report of the findings using R Markdown
- This course will work with two datasets from the ILO (International Labor Organization)
- ilo_working_hours - country-year-working_hours
- ilo_hourly_compensation - coutry-year-hourly_compensation
- ilo_working_hours - country-year-working_hours
- Begin by integrating the data using dplyr::inner_join
- Usually, categorical variables like country in this example should be converted to factors before plotting them
- You can do so using as.factor(). In your data set, two columns are still of type “character” – use mutate() to turn them into factors
Filtering and plotting the data:
- The filter() function can be used to maintain only the European countries - best for the key years of interest
- The %in% operator will be valuable for this, given a vector of countries in Europe
- The %in% operator will be valuable for this, given a vector of countries in Europe
- Will look at histograms, scatter-plots, titling, and the like
- Will use group_by() and summarize() also for looking at tabular results
Custom ggplot2 themes - providing a custom look to a chart:
- Custom looks can make it easier to highlight key data - colors, emphasis, shading, etc.
- The theme() function is added to a function just like anything else in a ggplot
- text=element_text(family=, color=) # to make a specific family and color available for all the labels and text
- text=element_text(family=, color=) # to make a specific family and color available for all the labels and text
- Can also add default ggplot2 themes to a plot
- Can chain themes, including a default theme followed by several overrides
- theme_classic() + theme(text=element_text(family=, color=)
- theme_classic() + theme(text=element_text(family=, color=)
- Can get an overview of all the possible options by using ?theme
- There are four key members of the element_* function family
- element_text()
- element_rect()
- element_line()
- element_blank() - makes plot elements disappear
- element_text()
Example code includes:
library(ggplot2)
load("./RInputFiles/ilo_hourly_compensation.RData")
load("./RInputFiles/ilo_working_hours.RData")
# Join both data frames
ilo_data <- ilo_hourly_compensation %>%
inner_join(ilo_working_hours, by = c("country", "year"))
# Count the resulting rows
ilo_data %>%
count()## # A tibble: 1 x 1
## n
## <int>
## 1 612# Examine ilo_data
ilo_data## # A tibble: 612 x 4
## country year hourly_compensation working_hours
## <chr> <chr> <dbl> <dbl>
## 1 Australia 1980.0 8.44 34.6
## 2 Canada 1980.0 8.87 34.8
## 3 Denmark 1980.0 10.8 31.9
## 4 Finland 1980.0 8.61 35.6
## 5 France 1980.0 8.90 35.4
## 6 Italy 1980.0 8.09 35.7
## 7 Japan 1980.0 5.46 40.8
## 8 Korea, Rep. 1980.0 0.950 55.3
## 9 Norway 1980.0 11.8 30.4
## 10 Spain 1980.0 5.86 36.8
## # ... with 602 more rows# Turn year into a factor
ilo_data <- ilo_data %>%
mutate(year = as.factor(as.numeric(year)))
# Turn country into a factor
ilo_data <- ilo_data %>%
mutate(country = as.factor(country))
# Examine the European countries vector
european_countries <- c('Finland', 'France', 'Italy', 'Norway', 'Spain', 'Sweden', 'Switzerland', 'United Kingdom', 'Belgium', 'Ireland', 'Luxembourg', 'Portugal', 'Netherlands', 'Germany', 'Hungary', 'Austria', 'Czech Rep.')
european_countries## [1] "Finland" "France" "Italy" "Norway"
## [5] "Spain" "Sweden" "Switzerland" "United Kingdom"
## [9] "Belgium" "Ireland" "Luxembourg" "Portugal"
## [13] "Netherlands" "Germany" "Hungary" "Austria"
## [17] "Czech Rep."# Only retain European countries
ilo_data <- ilo_data %>%
filter(country %in% european_countries)
# Examine the structure of ilo_data
str(ilo_data)## Classes 'tbl_df', 'tbl' and 'data.frame': 380 obs. of 4 variables:
## $ country : Factor w/ 30 levels "Australia","Austria",..: 8 9 15 22 25 27 28 29 8 9 ...
## $ year : Factor w/ 27 levels "1980","1981",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ hourly_compensation: num 8.61 8.9 8.09 11.8 5.86 ...
## $ working_hours : num 35.6 35.4 35.7 30.4 36.8 ...# Group and summarize the data
ilo_data %>%
group_by(year) %>%
summarize(mean_hourly_compensation = mean(hourly_compensation),
mean_working_hours = mean(working_hours))## # A tibble: 27 x 3
## year mean_hourly_compensation mean_working_hours
## <fct> <dbl> <dbl>
## 1 1980 9.27 34.0
## 2 1981 8.69 33.6
## 3 1982 8.36 33.5
## 4 1983 7.81 33.9
## 5 1984 7.54 33.7
## 6 1985 7.79 33.7
## 7 1986 9.70 34.0
## 8 1987 12.1 33.6
## 9 1988 13.2 33.7
## 10 1989 13.1 33.5
## # ... with 17 more rows# Filter for 2006
plot_data <- ilo_data %>%
filter(year == 2006)
# Create the scatter plot
ggplot(plot_data) +
geom_point(aes(x = working_hours, y = hourly_compensation))
# Create the plot
ggplot(plot_data) +
geom_point(aes(x = working_hours, y = hourly_compensation)) +
# Add labels
labs(
x = "Working hours per week",
y = "Hourly compensation",
title = "The more people work, the less compensation they seem to receive",
subtitle = "Working hours and hourly compensation in European countries, 2006",
caption = "Data source: ILO, 2017"
)
# Save your current plot into a variable: ilo_plot
ilo_plot <- ggplot(plot_data) +
geom_point(aes(x = working_hours, y = hourly_compensation)) +
labs(
x = "Working hours per week",
y = "Hourly compensation",
title = "The more people work, the less compensation they seem to receive",
subtitle = "Working hours and hourly compensation in European countries, 2006",
caption = "Data source: ILO, 2017"
)
# Try out theme_minimal
ilo_plot +
theme_minimal()
# Try out any other possible theme function
ilo_plot +
theme_linedraw()
windowsFonts(Bookman=windowsFont("Bookman Old Style"))
ilo_plot <- ilo_plot +
theme_minimal() +
# Customize the "minimal" theme with another custom "theme" call
theme(
text = element_text(family = "Bookman"),
title = element_text(color = "gray25"),
plot.subtitle = element_text(size=12),
plot.caption = element_text(color = "gray30")
)
# Render the plot object
ilo_plot
ilo_plot +
# "theme" calls can be stacked upon each other, so this is already the third call of "theme"
theme(
plot.background = element_rect(fill = "gray95"),
plot.margin = unit(c(5, 10, 5, 10), units = "mm")
)
Chapter 2 - Creating Custom and Unique Visualization
Visualizing aspects of data with facets:
- The facet_grid() function builds on the facet_wrap() concept, allowing for further control
- facet_grid(rowVar ~ colVar)
- Note that facet_grid(. ~ year) will give the same output as facet_wrap(~ year)
- facet_grid(rowVar ~ colVar)
- Theme options are available for faceted plots - strip.backgroumd, strip.text, etc.
- Can also create your own theme functions, such as:
- theme_green <- function(){ theme( plot.background = element_rect(fill = “green”), panel.background = element_rect(fill = “lightgreen”) ) }
Custom plot to emphasize change:
- The dot plot is useful for comparing change over time
- Dot for starting point, arrow pointint towards ending point, text labels at start and end of arrow, arranged so that country (or whatever) is along the y-axis
- Dot for starting point, arrow pointint towards ending point, text labels at start and end of arrow, arranged so that country (or whatever) is along the y-axis
- The default geom_dotplot() is NOT what is needed - this is a histogram using dots rather than bars
- Instead, the geom_path() is available for connecting observations in the order in which they appear in the data (so, proper ordering of the data frame is VERY important!)
- The geom_path(aes(x=, y=), arrow=arrow()) will expect at least one numeric variable, and one variable (y) that is either numeric or factor
- The arrow() is a function that allows for calling a specific type of arrow, arrow head, and the like
- The geom_path(aes(x=, y=), arrow=arrow()) will expect at least one numeric variable, and one variable (y) that is either numeric or factor
Polishing the dot plot:
- Ordering the factors can help make things much clearer in the ggplot - ggplot defaults to using the factor levels
- The library(forcats) is great for working with factor variables, and is part of the tidyverse
- fct_drop for dropping levels
- fct_rev for reversing factor levels
- fct_reorder for reordering factor levels
- fct_drop for dropping levels
- The arguments for fct_reorder(factorVar, dataVar, FUN) - frequently applied by way of a mutate() call
- Can further use the hjust and vjust aesthetics to nudge the labels for better readability
- These are added inside the aes() call for geom_text() and can be like aes(…, hjust=ifelse(year == 2006, 1.4, -0.4))
Finalizing plots for different audiences and devices:
- Changing the viewport (zooming or repositioning) can be managed in any of two manners
- coord_cartesian(xlim=c(), ylim=c()) is the default ggplot2 mechanism
- The difference with using coord_cartesian rather than direct +xlim() + ylim() is that coord_cartesian() will prevent clipping, which is generally preferred
- coord_cartesian(xlim=c(), ylim=c()) is the default ggplot2 mechanism
- Need to customize the plot for mobile devices
- Can be helpful to have the plot available in 16:9 aspect ratio, which nicely fits most smartphones
- Can also be helpful to kill off axes, and put any labels needed directly in to the data
- Can be helpful to have the plot available in 16:9 aspect ratio, which nicely fits most smartphones
- In this exercise, you’re going to encounter something that is probably new to you
- New data sets can be given to single geometries like geom_text(), so these geometries don’t use the data set given to the initial ggplot() call
- In this exercise, you are going to need this because you only want to add one label to each arrow
- If you were to use the original data set ilo_data, two labels would be added because there are two observations for each country in the data set, one for 1996 and one for 2006
- New data sets can be given to single geometries like geom_text(), so these geometries don’t use the data set given to the initial ggplot() call
Example code includes:
# Filter ilo_data to retain the years 1996 and 1996
ilo_data <- ilo_data %>%
filter(year == 1996 | year == 2006)
# Again, you save the plot object into a variable so you can save typing later on
ilo_plot <- ggplot(ilo_data, aes(x = working_hours, y = hourly_compensation)) +
geom_point() +
labs(
x = "Working hours per week",
y = "Hourly compensation",
title = "The more people work, the less compensation they seem to receive",
subtitle = "Working hours and hourly compensation in European countries, 2006",
caption = "Data source: ILO, 2017"
) +
# Add facets here
facet_grid(facets = . ~ year)
ilo_plot
# For a starter, let's look at what you did before: adding various theme calls to your plot object
ilo_plot +
theme_minimal() +
theme(
text = element_text(family = "Bookman", color = "gray25"),
plot.subtitle = element_text(size = 12),
plot.caption = element_text(color = "gray30"),
plot.background = element_rect(fill = "gray95"),
plot.margin = unit(c(5, 10, 5, 10), units = "mm")
)
# Define your own theme function below
theme_ilo <- function() {
theme_minimal() +
theme(
text = element_text(family = "Bookman", color = "gray25"),
plot.subtitle = element_text(size = 12),
plot.caption = element_text(color = "gray30"),
plot.background = element_rect(fill = "gray95"),
plot.margin = unit(c(5, 10, 5, 10), units = "mm"))
}
# Apply your theme function
ilo_plot <- ilo_plot + theme_ilo()
# Examine ilo_plot
ilo_plot
ilo_plot +
# Add another theme call
theme(
# Change the background fill to make it a bit darker
strip.background = element_rect(fill = "gray60", color = "gray95"),
# Make text a bit bigger and change its color to white
strip.text = element_text(size = 11, color = "white")
)
# Create the dot plot
ggplot(ilo_data) +
geom_path(aes(x=working_hours, y=country))
ggplot(ilo_data) +
geom_path(aes(x = working_hours, y = country),
# Add an arrow to each path
arrow = arrow(length = unit(1.5, "mm"), type = "closed"))
ggplot(ilo_data) +
geom_path(aes(x = working_hours, y = country),
arrow = arrow(length = unit(1.5, "mm"), type = "closed")) +
# Add a geom_text() geometry
geom_text(
aes(x = working_hours,
y = country,
label = round(working_hours, 1))
)
library(forcats)
# Reorder country factor levels
ilo_data <- ilo_data %>%
# Arrange data frame
arrange(country, year) %>%
# Reorder countries by working hours in 2006
mutate(country = fct_reorder(country,
working_hours,
last))
# Plot again
ggplot(ilo_data) +
geom_path(aes(x = working_hours, y = country),
arrow = arrow(length = unit(1.5, "mm"), type = "closed")) +
geom_text(
aes(x = working_hours,
y = country,
label = round(working_hours, 1))
)
# Save plot into an object for reuse
ilo_dot_plot <- ggplot(ilo_data) +
geom_path(aes(x = working_hours, y = country),
arrow = arrow(length = unit(1.5, "mm"), type = "closed")) +
# Specify the hjust aesthetic with a conditional value
geom_text(
aes(x = working_hours,
y = country,
label = round(working_hours, 1),
hjust = ifelse(year == "2006", 1.4, -0.4)
),
# Change the appearance of the text
size = 3,
family = "Bookman",
color = "gray25"
)
ilo_dot_plot
# Reuse ilo_dot_plot
ilo_dot_plot <- ilo_dot_plot +
# Add labels to the plot
labs(
x = "Working hours per week",
y = "Country",
title = "People work less in 2006 compared to 1996",
subtitle = "Working hours in European countries, development since 1996",
caption = "Data source: ILO, 2017"
) +
# Apply your theme
theme_ilo() +
# Change the viewport
coord_cartesian(xlim = c(25, 41))
# View the plot
ilo_dot_plot
# Compute temporary data set for optimal label placement
median_working_hours <- ilo_data %>%
group_by(country) %>%
summarize(median_working_hours_per_country = median(working_hours)) %>%
ungroup()
# Have a look at the structure of this data set
str(median_working_hours)## Classes 'tbl_df', 'tbl' and 'data.frame': 17 obs. of 2 variables:
## $ country : Factor w/ 30 levels "Netherlands",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ median_working_hours_per_country: num 27 27.8 28.4 31 30.9 ...ilo_dot_plot +
# Add label for country
geom_text(data = median_working_hours,
aes(y = country,
x = median_working_hours_per_country,
label = country),
vjust = -0.5,
size=3,
family = "Bookman",
color = "gray25") +
# Remove axes and grids
theme(
axis.ticks = element_blank(),
axis.title = element_blank(),
axis.text = element_blank(),
panel.grid = element_blank(),
# Also, let's reduce the font size of the subtitle
plot.subtitle = element_text(size = 9)
)
Chapter 3 - Introduction to R Markdown
What is R Markdown?
- R Markdown is a framework for converting R code in to a wide range of outputs - html, PDF, etc.
- R Markdown -> knitr -> md -> pandoc -> final output
- R Markdown -> knitr -> md -> pandoc -> final output
- The biggest benefit of R Markdown is the full reproducibility of the analysis
- Other people or yourself (if there is new data)
- Other people or yourself (if there is new data)
- The code needs to be executable on other people’s machines, and the data should either be included (or have a link to it)
- These are the minimum standards for reproducibility
- Fuller standards would include software/package systems, run date/time, operating system, etc.
- These are the minimum standards for reproducibility
Formatting with R Markdown:
- Markdown is an example of a mark-up language (like html, which is a hyper-text mark-up language)
- Markdown was created to have a quick means of creating valid html code that could be published on the internet
- Markdown was created to have a quick means of creating valid html code that could be published on the internet
- The hash signs (#) are for levels of header - # (first), ## (second), etc.
- The asterisk will make italics with singles (myItalicText) and bold with doubles (myBoldText)
- Links can be introduced and named using the target name inside square brackets followed by the target link inside parentheses myLinkName
- The R Markdown document is a mix of R code and Markdown code
- R Markdown uses pandoc markdown, part of the markdown family
- Can use the pandoc markdown reference guide, available at R Studio
- R Markdown uses pandoc markdown, part of the markdown family
R Code in R Markdown Documents:
- Adding R chunks is as simple as adding triple back-ticks, followed by {r}, ended with triple back-ticks
- Can also add R code inline such as back-tick r 2 + 2 back-tick, such as 2 + 2 equals 4
- There are many options available for R code chunks including
- include=FALSE # execute the R code but do not quote it or print any output
- echo=FALSE # do not show the R code, but show its output
- message=FALSE # do not show messages
- warning=FALSE # do not show warnings
- eval=FALSE # do not evaluate the R chunk (but do print it provided the default echo=TRUE is set)
- include=FALSE # execute the R code but do not quote it or print any output
- R code chunks can also be named
- This makes sense in large documents, especially if there is an error inside a chunk
- While knitting a document in RStudio, for example, the error can then be located in an easier fashion
- Chunk options are added after the name of the chunk and a comma, like so: {r name, option=value}
- This makes sense in large documents, especially if there is an error inside a chunk
Images in R Markdown Files:
- Images resulting from code are responsive, which is to say that they will change with the page size
- Sometimes, the default options that go with a figure are sub-optimal (wrong aspect ratio or the like)
- Can add options like fig.height=6 inside the ```{r} command to address these - default unit is inches
- Also can use fig.width= (inches) and fig.align= (any of “right”, “left”, “center”)
- Sometimes, the default options that go with a figure are sub-optimal (wrong aspect ratio or the like)
- Can also load external images in to Markdown
- exclamation-mark square-brackets-containing-name parentheses-containing-image-location
Example code is contained in the summary Excel worksheet.
Chapter 4 - Customizing R Markdown Reports
Advanced YAML Settings (YAML is a recursive name meaning YAML and Markup Language):
- YAML documents typically start and end with three hyphens (—) with value: key pairs
- Indentations suggest sub-family relationships; spacing does not matter, but everything of the same level must be indented the same
- Indentations suggest sub-family relationships; spacing does not matter, but everything of the same level must be indented the same
- All R Markdown documents begin with a YAML header, which can be customized and enhanced
- You add a table with toc: true and specify whether it should be floating (= whether it moves along as you scroll) with toc_float
- The depth of your TOC is set with toc_depth
- The depth of your TOC is set with toc_depth
- Before you dig deeper into custom stylesheets, let’s enable code folding with code_folding: ….
- This will allow your readers to completely hide all code chunks or show them – all at once or individually
Custom stylesheets - creating a unique theme for a report:
- Can refer to any CSS (cascading style sheet) in the YAML header
- Can use any of the html tags that would be created by the document, and enhance the properties they will have for this html document output
- h2 { font-family: “Bookman”, serif; }
- Conclude each rule with a semicolon!
- body, h1, h2 { font-family: “Bookman”, serif; }
- Separate the html tags with commas
- a { color: #0000FF; font-weight: bold; }
- Separate the commands with a semicolon (same as used to end the commands)
- h2 { font-family: “Bookman”, serif; }
- There are some further customization possibilities
- strong { color: “blue”; } # this will make everything of tag “strong” blue
- strong.red { color: “red”; } # can create a strong.red tag that will be “red” even while the rest of them are “blue”
- strong { color: “blue”; } # this will make everything of tag “strong” blue
- It is also possible to specify combinators in CSS, which is to say that tags within another tag only are impacted
- div strong { color: “green”; } # strong tags subordinate at any number of levels to div tags will be colored green
- div > strong { color: “red”; } # strong tags directly subordinate (pure child-parent relationship) to div will be colored red
- div strong { color: “green”; } # strong tags subordinate at any number of levels to div tags will be colored green
- Mozilla Developer Network has a lot of style tag ideas
Beautiful tables:
- By default, R Markdown renders tables exactly as they would be rendered to the R console
- Can add the df_print: key-value under the html_document: (or whatever) area
- Typically key-value are either df_print: kable or df_print: paged
- Typically key-value are either df_print: kable or df_print: paged
- Alternately, to just change a single table, pipe the output to knitr::kable()
- myData %>% group_by(myFactor) %>% summarize(myEquations) %>% knitr::kable()
- myData %>% group_by(myFactor) %>% summarize(myEquations) %>% knitr::kable()
- Tables can also be styled using html tags - basic anatomy of a table includes
… …
- Each of the header and the body will have one or more rows, each depicted using
… - Each row of the header is
Column1 Column 2 . . .
- Each row of the body is
Cell1 Cell 2 . . .
- Add %>% pull(n) (from dplyr) to the inline R statement in the “Data” section, so its output is not rendered as a table
- pull() extracts single columns out of data frames
Summary:
- Course summarized the final component of the tidyverse process - communication is key!
- Switzerland demographic map
- Can show population density using geom_line()
Example code is contained in the summary Excel worksheet.
Foundations of Probability in R
Chapter 1 - Binomial Distribution
Flipping coins in R - for example, rbinom(1, 1, 0.5) - 1 draw of 1 coint with 50% of being heads:
- Generally, interpretation of 1 is heads
- rbinom(nDraws, nPerDraw, pPerDraw) - can generate multiple simulations at the same time
- Frequent focus in this course will be on biased coins - pPerDraw != 0.5
Density and cumulative density:
- Histogram on a simulation can be a helpful way for understanding densities and likelihoods
- With a known distribution, can get the exact answer using dbinom(nHit, nDraw, pPerDraw)
- The “cumulative density” is the probability of getting this value or less
- pbinom(nHit, nDraw, pPerDraw) # gives the cumulative probability of nHit or ferwer hits when making nDraw draws each at probability pPerDraw
Expected value and variance:
- Two interesting characteristics are the expected value and the variance of the distribution
- The theoretical mean for the binomial is easy to calculate from the parameters
- mean = n * p
- mean = n * p
- The theroretical variance (mean-squared distance from the mean) for the binomial is also easy to calculate from the parameters
- variance = n * p * (1 - p)
Example code includes:
# Generate 10 separate random flips with probability .3 rbinom(10, 1, 0.3)## [1] 0 1 0 0 1 1 0 0 1 0# Generate 100 occurrences of flipping 10 coins, each with 30% probability rbinom(100, 10, 0.3)## [1] 2 1 6 4 2 3 3 6 8 5 1 1 3 7 1 5 4 6 4 3 4 2 4 2 4 1 2 5 1 7 2 5 2 5 3 ## [36] 4 5 2 3 3 0 4 3 3 5 2 4 1 2 3 2 1 4 5 4 0 5 6 5 2 1 2 3 2 2 4 2 5 3 5 ## [71] 3 4 1 2 4 1 3 2 6 3 4 2 4 6 6 2 2 2 4 6 4 4 2 1 4 3 0 4 3 3# Calculate the probability that 2 are heads using dbinom dbinom(2, 10, 0.3)## [1] 0.2334744# Confirm your answer with a simulation using rbinom mean(rbinom(10000, 10, 0.3) == 2)## [1] 0.2353# Calculate the probability that at least five coins are heads 1 - pbinom(4, 10, 0.3)## [1] 0.1502683# Confirm your answer with a simulation of 10,000 trials mean(rbinom(10000, 10, 0.3) >= 5)## [1] 0.1533# Here is how you computed the answer in the last problem mean(rbinom(10000, 10, .3) >= 5)## [1] 0.149# Try now with 100, 1000, 10,000, and 100,000 trials mean(rbinom(100, 10, .3) >= 5)## [1] 0.16mean(rbinom(1000, 10, .3) >= 5)## [1] 0.158mean(rbinom(10000, 10, .3) >= 5)## [1] 0.1518mean(rbinom(100000, 10, .3) >= 5)## [1] 0.15187# Calculate the expected value using the exact formula 25 * 0.3## [1] 7.5# Confirm with a simulation using rbinom mean(rbinom(10000, 25, 0.3))## [1] 7.4447# Calculate the variance using the exact formula 25 * 0.3 * (1 - 0.3)## [1] 5.25# Confirm with a simulation using rbinom var(rbinom(10000, 25, 0.3))## [1] 5.15845
Chapter 2 - Laws of Probability
Probability of Event A and Event B:
- Suppose there are two independent events, possibly with different probabilities, A and B
- P(A and B) = P(A) * P(B) # assuming A and B are independent, as assumed throughout this chapter
- P(A and B) = P(A) * P(B) # assuming A and B are independent, as assumed throughout this chapter
- If there are two boolean vectors, A and B, then A & B will give a single boolean vector that is the “and” on each pair of elements
Probability of A or B:
- P(A or B) = P(A) + P(B) - P(A and B)
- Alternately, P(A or B) = 1 - P(notA and notB)
- Can also use the mean(A | B) assuming that A and B are boolean vectors of the same length
- Alternately, P(A or B) = 1 - P(notA and notB)
Multiplying random variables:
- Suppose that you already have a variable X with a known mean and variance
- mean(a * X) = a * mean(X)
- var(a * X) = a^2 * var(X)
- mean(a * X) = a * mean(X)
Adding random variables:
- Suppose that you already have random variables X and Y with known means and variances
- mean(X + Y) = mean(X) + mean(Y) # does not require independence
- var(X + Y) = var(X) + var(Y) # requires independence
- mean(X + Y) = mean(X) + mean(Y) # does not require independence
Example code includes:
# Simulate 100,000 flips of a coin with a 40% chance of heads A <- rbinom(100000, 1, 0.4) # Simulate 100,000 flips of a coin with a 20% chance of heads B <- rbinom(100000, 1, 0.2) # Estimate the probability both A and B are heads mean(A & B)## [1] 0.0805# You've already simulated 100,000 flips of coins A and B A <- rbinom(100000, 1, .4) B <- rbinom(100000, 1, .2) # Simulate 100,000 flips of coin C (70% chance of heads) C <- rbinom(100000, 1, .7) # Estimate the probability A, B, and C are all heads mean(A & B & C)## [1] 0.05589# Simulate 100,000 flips of a coin with a 60% chance of heads A <- rbinom(100000, 1, 0.6) # Simulate 100,000 flips of a coin with a 10% chance of heads B <- rbinom(100000, 1, 0.1) # Estimate the probability either A or B is heads mean(A | B)## [1] 0.63736# Use rbinom to simulate 100,000 draws from each of X and Y X <- rbinom(100000, 10, 0.6) Y <- rbinom(100000, 10, 0.7) # Estimate the probability either X or Y is <= to 4 mean((X <= 4) | (Y <= 4))## [1] 0.20613# Use pbinom to calculate the probabilities separately prob_X_less <- pbinom(4, 10, 0.6) prob_Y_less <- pbinom(4, 10, 0.7) # Combine these to calculate the exact probability either <= 4 prob_X_less + prob_Y_less - prob_X_less * prob_Y_less## [1] 0.2057164# Simulate 100,000 draws of a binomial with size 20 and p = .1 X <- rbinom(100000, 20, 0.1) # Estimate the expected value of X mean(X)## [1] 1.9991# Estimate the expected value of 5 * X mean(5 * X)## [1] 9.9955# Estimate the variance of X var(X)## [1] 1.786197# Estimate the variance of 5 * X var(5 * X)## [1] 44.65493# Simulate 100,000 draws of X (size 20, p = .3) and Y (size 40, p = .1) X <- rbinom(100000, 20, 0.3) Y <- rbinom(100000, 40, 0.1) # Estimate the expected value of X + Y mean(X + Y)## [1] 9.99048# Find the variance of X + Y var(X + Y)## [1] 7.798627# Find the variance of 3 * X + Y var(3 * X + Y)## [1] 41.20331
Chapter 3 - Bayesian Statistics
Updating with evidence:
- Probability of A given B -> P(A | B) = P(A and B) / P(B)
Prior probability - may not be equal odds prior to seeing any evidence:
- The prior probability is the belief in the probabilities prior to seeing any evidence
- Can just simulate the relative sizes - for example, if there is a 9:1 prior, simulate 90,000 vs. 10,000 before finding conditional probability
Bayes theorem:
- Basically multiply prior probability for A with likelihood of seeing event (density) if A
- Repeat for B, C, …
- Scale multiplied probabilities to add to one, and those are the posterior probabilities
- Pr(A|B) = P(A and B) / P(B)
- Repeat for B, C, …
- The more generalized Bayes theory is
- Numer = P(B|A) * P(A)
- Denom = P(B|A) * P(A) + P(B | notA) * P(notA)
- P(A|B) = Numer / Denom
- Numer = P(B|A) * P(A)
Example code includes:
# Simulate 50000 cases of flipping 20 coins from fair and from biased fair <- rbinom(50000, 20, 0.5) biased <- rbinom(50000, 20, 0.75) # How many fair cases, and how many biased, led to exactly 11 heads? fair_11 <- sum(fair == 11) biased_11 <- sum(biased == 11) # Find the fraction of fair coins that are 11 out of all coins that were 11 fair_11 / (fair_11 + biased_11)## [1] 0.8487457# How many fair cases, and how many biased, led to exactly 16 heads? fair_16 <- sum(fair == 16) biased_16 <- sum(biased == 16) # Find the fraction of fair coins that are 16 out of all coins that were 16 fair_16 / (fair_16 + biased_16)## [1] 0.02418033# Simulate 8000 cases of flipping a fair coin, and 2000 of a biased coin fair_flips <- rbinom(8000, 20, 0.5) biased_flips <- rbinom(2000, 20, 0.75) # Find the number of cases from each coin that resulted in 14/20 fair_14 <- sum(fair_flips == 14) biased_14 <- sum(biased_flips == 14) # Use these to estimate the posterior probability fair_14 / (fair_14 + biased_14)## [1] 0.4651515# Simulate 80,000 draws from fair coin, 10,000 from each of high and low coins flips_fair <- rbinom(80000, 20, 0.5) flips_high <- rbinom(10000, 20, 0.75) flips_low <- rbinom(10000, 20, 0.25) # Compute the number of coins that resulted in 14 heads from each of these piles fair_14 <- sum(flips_fair == 14) high_14 <- sum(flips_high == 14) low_14 <- sum(flips_low == 14) # Compute the posterior probability that the coin was fair fair_14 / (fair_14 + high_14 + low_14)## [1] 0.6370197# Use dbinom to calculate the probability of 11/20 heads with fair or biased coin probability_fair <- dbinom(11, 20, 0.5) probability_biased <- dbinom(11, 20, 0.75) # Calculate the posterior probability that the coin is fair probability_fair / (probability_fair + probability_biased)## [1] 0.8554755# Find the probability that a coin resulting in 14/20 is fair probability_fair <- dbinom(14, 20, .5) probability_biased <- dbinom(14, 20, .75) probability_fair / (probability_fair + probability_biased)## [1] 0.179811# Find the probability that a coin resulting in 18/20 is fair probability_fair <- dbinom(18, 20, .5) probability_biased <- dbinom(18, 20, .75) probability_fair / (probability_fair + probability_biased)## [1] 0.002699252# Use dbinom to find the probability of 16/20 from a fair or biased coin probability_16_fair <- dbinom(16, 20, 0.5) probability_16_biased <- dbinom(16, 20, 0.75) # Use Bayes' theorem to find the posterior probability that the coin is fair (probability_16_fair * 0.99) / (probability_16_fair * 0.99 + probability_16_biased * 0.01)## [1] 0.7068775
Chapter 4 - Related Distributions
Normal distribution - symmetrical bell curve, Gaussian:
- The normal distribution can be defined by mean and standard deviation (or mean and variance)
- Can simulate from the normal distribution with rnorm(n, mean, sd)
Poisson distribution - approximates the binomial under the assumption of a large number of trials each with a low probability:
- The Poisson distribution is described only by its mean, lambda
- Basically, lambda is nDraw * pPerDraw
- The variance of the Poisson distribution is equal to the mean
- Basically, lambda is nDraw * pPerDraw
- The Poisson distribution is best for modeling rare events where you really just care about counts (not proportions of a total potential universe)
- One of the useful properties of the Poisson distribution is that when you add multiple Poisson distributions together, the result is also a Poisson distribution
Geometric distribution - example of flipping a coin with probability p and assessing when the first success occurs:
- The replicate() function is basically a wrapper to sapply() and can be helpful for simulations like this
- replicate(10, which(binom(100, 1, 0.1) == 1)[1])
- replicate(10, which(binom(100, 1, 0.1) == 1)[1])
- The rgeom(nDraws, prob) will give back the geometric distribution
- The mean will be 1/prob - 1 since it is is the number of trials “before” the first success
- The mean would be 1/prob if instead the question is the number of trials to get the first success
- The mean will be 1/prob - 1 since it is is the number of trials “before” the first success
Example code includes:
compare_histograms <- function(variable1, variable2) { x <- data.frame(value = variable1, variable = "Variable 1") y <- data.frame(value = variable2, variable = "Variable 2") ggplot(rbind(x, y), aes(value)) + geom_histogram() + facet_wrap(~ variable, nrow = 2) } # Draw a random sample of 100,000 from the Binomial(1000, .2) distribution binom_sample <- rbinom(100000, 1000, 0.2) # Draw a random sample of 100,000 from the normal approximation normal_sample <- rnorm(100000, 200, sqrt(160)) # Compare the two distributions with the compare_histograms function compare_histograms(binom_sample, normal_sample)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Use binom_sample to estimate the probability of <= 190 heads mean(binom_sample <= 190)## [1] 0.2292# Use normal_sample to estimate the probability of <= 190 heads mean(normal_sample <= 190)## [1] 0.2152# Calculate the probability of <= 190 heads with pbinom pbinom(190, 1000, 0.2)## [1] 0.2273564# Calculate the probability of <= 190 heads with pnorm pnorm(190, 200, sqrt(160))## [1] 0.2145977# Draw a random sample of 100,000 from the Binomial(10, .2) distribution binom_sample <- rbinom(100000, 10, 0.2) # Draw a random sample of 100,000 from the normal approximation normal_sample <- rnorm(100000, 2, sqrt(1.6)) # Compare the two distributions with the compare_histograms function compare_histograms(binom_sample, normal_sample)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Draw a random sample of 100,000 from the Binomial(1000, .002) distribution binom_sample <- rbinom(100000, 1000, 0.002) # Draw a random sample of 100,000 from the Poisson approximation poisson_sample <- rpois(100000, 2) # Compare the two distributions with the compare_histograms function compare_histograms(binom_sample, poisson_sample)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Find the percentage of simulated values that are 0 mean(poisson_sample == 0)## [1] 0.13513# Use dpois to find the exact probability that a draw is 0 dpois(0, 2)## [1] 0.1353353# Simulate 100,000 draws from Poisson(1) X <- rpois(100000, 1) # Simulate 100,000 draws from Poisson(2) Y <- rpois(100000, 2) # Add X and Y together to create Z Z <- X + Y # Use compare_histograms to compare Z to the Poisson(3) compare_histograms(Z, rpois(100000, 3))## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Simulate 100 instances of flipping a 20% coin flips <- rbinom(100, 1, 0.2) # Use which to find the first case of 1 ("heads") which(flips == 1)[1]## [1] 6# Existing code for finding the first instance of heads which(rbinom(100, 1, .2) == 1)[1]## [1] 5# Replicate this 100,000 times using replicate() replications <- replicate(100000, which(rbinom(100, 1, .2) == 1)[1]) # Histogram the replications with qplot qplot(replications)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Generate 100,000 draws from the corresponding geometric distribution geom_sample <- rgeom(100000, 0.2) # Compare the two distributions with compare_histograms compare_histograms(replications, geom_sample)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Find the probability the machine breaks on 5th day or earlier pgeom(4, 0.1)## [1] 0.40951# Find the probability the machine is still working on 20th day 1 - pgeom(19, 0.1)## [1] 0.1215767# Calculate the probability of machine working on day 1-30 still_working <- 1 - pgeom(0:29, 0.1) # Plot the probability for days 1 to 30 qplot(1:30, still_working)
Inference for Numerical Data
Chapter 1 - Bootstrapping for Parameter Estimates
Introduction - beginning with bootstrapping approach:
- Example of 20 random apartment rents available in Manhattan
- Median is the best statistic
- Median is the best statistic
- Bootstrap comes from the phrase “pulling yourself up by the bootstraps” (doing the impossible without any help)
- Take many random samples with replacement of the same length as the sample data, take their medians, and find the summary statistics about the median
- The bootstrap distribution is like multiple samples from the sample population
- Take many random samples with replacement of the same length as the sample data, take their medians, and find the summary statistics about the median
- Can run bootstraps from the infer package, for example
- myData %>% infer::specify(response=) %>% infer::generate(reps=, type=“bootstrap”) %>% infer::calculate(stat=“”)
Percentile and standard error methods:
- Sampling with replacement allows for each item in the sample to potentially be in the population many more times
- Can describe a bootstrap statistic using a CI, such as the 95th percentile
- A more accurate calculation is typically to use the standard error approach
- sample statistic +/- t(df=n-1) * SEboot
Re-centering bootstrap distributions for hypothesis testing:
- Simulation methods to test whether a bootstrap parameter is less than, different than, or greater than a critical value
- There is a multi-step process that includes
- Bootstrap distribution is centered around the same statistics to begin with
- Since we are now assuming Ho to be true, we shift the bootstrap distribution right/left as needed so that this default is true
- The p-value is then the number of observations that are at least as favorable to the alternate hypothesis as the observed sample statistic
- Bootstrap distribution is centered around the same statistics to begin with
Example code includes:
manhattan <- readr::read_csv("./RInputFiles/manhattan.csv")## Parsed with column specification: ## cols( ## rent = col_integer() ## )# Will need to either call library(infer) or add infer:: to this code library(infer) # Generate bootstrap distribution of medians rent_ci_med <- manhattan %>% # Specify the variable of interest specify(response = rent) %>% # Generate 15000 bootstrap samples generate(reps = 15000, type = "bootstrap") %>% # Calculate the median of each bootstrap sample calculate(stat = "median") # View the structure of rent_ci_med str(rent_ci_med)## Classes 'tbl_df', 'tbl' and 'data.frame': 15000 obs. of 2 variables: ## $ replicate: int 1 2 3 4 5 6 7 8 9 10 ... ## $ stat : num 2422 2350 2262 2325 2350 ... ## - attr(*, "response")= symbol rent# Plot a histogram of rent_ci_med ggplot(rent_ci_med, aes(x=stat)) + geom_histogram(binwidth=50)
# Percentile method rent_ci_med %>% summarize(l = quantile(stat, 0.025), u = quantile(stat, 0.975))## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 2162 2875# Standard error method # Calculate observed median rent_med_obs <- manhattan %>% # Calculate observed median rent summarize(median(rent)) %>% # Extract numerical value pull() # Determine critical value t_star <- qt(0.975, df = nrow(manhattan) - 1) # Construct interval rent_ci_med %>% summarize(boot_se = sd(rent_ci_med$stat)) %>% summarize(l = rent_med_obs - t_star * boot_se, u = rent_med_obs + t_star * boot_se)## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 1994 2706data(ncbirths, package="openintro") str(ncbirths)## 'data.frame': 1000 obs. of 13 variables: ## $ fage : int NA NA 19 21 NA NA 18 17 NA 20 ... ## $ mage : int 13 14 15 15 15 15 15 15 16 16 ... ## $ mature : Factor w/ 2 levels "mature mom","younger mom": 2 2 2 2 2 2 2 2 2 2 ... ## $ weeks : int 39 42 37 41 39 38 37 35 38 37 ... ## $ premie : Factor w/ 2 levels "full term","premie": 1 1 1 1 1 1 1 2 1 1 ... ## $ visits : int 10 15 11 6 9 19 12 5 9 13 ... ## $ marital : Factor w/ 2 levels "married","not married": 1 1 1 1 1 1 1 1 1 1 ... ## $ gained : int 38 20 38 34 27 22 76 15 NA 52 ... ## $ weight : num 7.63 7.88 6.63 8 6.38 5.38 8.44 4.69 8.81 6.94 ... ## $ lowbirthweight: Factor w/ 2 levels "low","not low": 2 2 2 2 2 1 2 1 2 2 ... ## $ gender : Factor w/ 2 levels "female","male": 2 2 1 2 1 2 2 2 2 1 ... ## $ habit : Factor w/ 2 levels "nonsmoker","smoker": 1 1 1 1 1 1 1 1 1 1 ... ## $ whitemom : Factor w/ 2 levels "not white","white": 1 1 2 2 1 1 1 1 2 2 ...# Remove NA visits ncbirths_complete_visits <- ncbirths %>% filter(!is.na(visits)) # Generate 15000 bootstrap means visit_ci_mean <- ncbirths_complete_visits %>% specify(response=visits) %>% generate(reps=15000, type="bootstrap") %>% calculate(stat="mean") # Calculate the 90% CI via percentile method visit_ci_mean %>% summarize(l = quantile(stat, 0.05), u = quantile(stat, 0.95))## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 11.9 12.3# Calculate 15000 bootstrap SDs visit_ci_sd <- ncbirths_complete_visits %>% specify(response=visits) %>% generate(reps=15000, type="bootstrap") %>% calculate(stat="sd") # Calculate the 90% CI via percentile method visit_ci_sd %>% summarize(l = quantile(stat, 0.05), u = quantile(stat, 0.95))## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 3.74 4.16# Generate 15000 bootstrap samples centered at null rent_med_ht <- manhattan %>% specify(response = rent) %>% hypothesize(null = "point", med = 2500) %>% generate(reps = 15000, type = "bootstrap") %>% calculate(stat = "median") # Calculate observed median rent_med_obs <- manhattan %>% summarize(median(rent)) %>% pull() # Calculate p-value rent_med_ht %>% filter(stat > rent_med_obs) %>% summarize(n() / 15000)## # A tibble: 1 x 1 ## `n()/15000` ## <dbl> ## 1 0.948# Generate 1500 bootstrap means centered at null weight_mean_ht <- ncbirths %>% specify(response = weight) %>% hypothesize(null = "point", mu = 7) %>% generate(reps=1500, type="bootstrap") %>% calculate(stat="mean") # Calculate observed mean weight_mean_obs <- ncbirths %>% summarize(mean(weight)) %>% pull() # Calculate p-value weight_mean_ht %>% filter(stat > weight_mean_obs) %>% summarize((n()/1500) * 2)## # A tibble: 1 x 1 ## `(n()/1500) * 2` ## <dbl> ## 1 0.0253
Chapter 2 - Introducing the t-distribution
The t-distribution - especially useful when the population standard deviation is unknown (as is typically the case):
- The t-distribution is like the normal distribution, but with thicker tails
- Observations are more likely to be 2+ SD from the mean using the t-distribution than with the normal distribution
- The t-distribution is always centered at zero, and has a single parameter, degrees of freedom
- Observations are more likely to be 2+ SD from the mean using the t-distribution than with the normal distribution
- As the degrees of freedom go to infinite, the t-distribution becomes the normal distribution
- Can always use the t-distribution, though
- Can always use the t-distribution, though
- We can use the pt function to find probabilities under the t-distribution
- For a given cutoff value q and a given degrees of freedom df, pt(q, df) gives us the probability under the t-distribution with df degrees of freedom for values of t less than q
- In other words, P(tdf<T)P(tdf<T) = pt(q = T, df)
- For a given cutoff value q and a given degrees of freedom df, pt(q, df) gives us the probability under the t-distribution with df degrees of freedom for values of t less than q
- We can use the qt() function to find cutoffs under the t-distribution
- For a given probability p and a given degrees of freedom df, qt(p, df) gives us the cutoff value for the t-distribution with df degrees of freedom for which the probability under the curve is p
- In other words, if P(tdf<T)=pP(tdf<T)=p, then TT = qt(p, df)
- For example, if TT corresponds to the 95th percentile of a distribution, p=0.95p=0.95
- For a given probability p and a given degrees of freedom df, qt(p, df) gives us the cutoff value for the t-distribution with df degrees of freedom for which the probability under the curve is p
Estimating a mean with a t-interval:
- Quantifying the expected variability of sample means - theory (CLM)
- The Central Limit Theorem (CLM) states that the sample mean will be normal with population mean and appropriate standard error (population sigma divided by sqrt(n) where n is the sample size)
- Since we do not have the original population, we never really have the population sigma
- However, the standard error is frequently estimated as the sample standard deviation divided by the square root of the sample size
- We use a t-distribution with df=n-1 to account for the extra uncertainty
- Since we do not have the original population, we never really have the population sigma
- The CLM has some key assumptions that must be validated first
- Independence of observations - hard to check, but typically assumed when the sampling methodology is appropriate
- The more skewed the original population, the larger sample size that is needed
- Independence of observations - hard to check, but typically assumed when the sampling methodology is appropriate
- The function t.test(myVar, conf.level=) will generate a confidence interval for the mean of myVar, as well as a p-value for the mean being non-zero
The t-interval for paired data:
- Examples would be same student taking two tests - this means the data are NOT independent, but instead they are paired
- Can be helpful in these cases to create a variable diff which is the difference in test scores by student
- Can then just run the normal t-test on the differences
- Can be helpful in these cases to create a variable diff which is the difference in test scores by student
Testing a mean with a t-test:
- Can run t.test(myVar, mu=myNullValue, alternative=“two.sided”) to run a two-sided t-test for mean(myVar) != myNullValue
- Will provide a p-value as well as a 95% CI for the mean of myVar
Example code includes:
# P(T < 3) for df = 10 (x <- pt(3, df = 10))## [1] 0.9933282# P(T > 3) for df = 10 (y <- 1 - pt(3, df=10))## [1] 0.006671828# P(T > 3) for df = 100 (z <- 1 - pt(3, df=100))## [1] 0.001703958# Comparison y == z## [1] FALSEy > z## [1] TRUEy < z## [1] FALSE# 95th percentile for df = 10 (x <- qt(0.95, df = 10))## [1] 1.812461# upper bound of middle 95th percent for df = 10 (y <- qt(0.975, df = 10))## [1] 2.228139# upper bound of middle 95th percent for df = 100 (z <- qt(0.975, df = 100))## [1] 1.983972# Comparison y == z## [1] FALSEy > z## [1] TRUEy < z## [1] FALSEdata(acs12, package="openintro") # Subset for employed respondents acs12_emp <- acs12 %>% filter(employment == "employed") # Construct 95% CI for avg time_to_work t.test(acs12_emp$time_to_work, conf.level=0.95)## ## One Sample t-test ## ## data: acs12_emp$time_to_work ## t = 32.635, df = 782, p-value < 2.2e-16 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 24.43369 27.56120 ## sample estimates: ## mean of x ## 25.99745t.test(acs12_emp$hrs_work, conf.level=0.95)## ## One Sample t-test ## ## data: acs12_emp$hrs_work ## t = 87.521, df = 842, p-value < 2.2e-16 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 38.05811 39.80429 ## sample estimates: ## mean of x ## 38.9312data(textbooks, package="openintro") # 90% CI t.test(textbooks$diff, conf.level = 0.9)## ## One Sample t-test ## ## data: textbooks$diff ## t = 7.6488, df = 72, p-value = 6.928e-11 ## alternative hypothesis: true mean is not equal to 0 ## 90 percent confidence interval: ## 9.981505 15.541783 ## sample estimates: ## mean of x ## 12.76164# 95% CI t.test(textbooks$diff, conf.level = 0.95)## ## One Sample t-test ## ## data: textbooks$diff ## t = 7.6488, df = 72, p-value = 6.928e-11 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 9.435636 16.087652 ## sample estimates: ## mean of x ## 12.76164# 99% CI t.test(textbooks$diff, conf.level = 0.99)## ## One Sample t-test ## ## data: textbooks$diff ## t = 7.6488, df = 72, p-value = 6.928e-11 ## alternative hypothesis: true mean is not equal to 0 ## 99 percent confidence interval: ## 8.347154 17.176133 ## sample estimates: ## mean of x ## 12.76164# Conduct HT t.test(textbooks$diff, mu=0, alternative="two.sided", conf.level=0.95)## ## One Sample t-test ## ## data: textbooks$diff ## t = 7.6488, df = 72, p-value = 6.928e-11 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 9.435636 16.087652 ## sample estimates: ## mean of x ## 12.76164# Calculate 15000 bootstrap means textdiff_med_ci <- textbooks %>% specify(response = diff) %>% generate(reps=15000, type="bootstrap") %>% calculate(stat = "median") # Calculate the 95% CI via percentile method textdiff_med_ci %>% summarize(l=quantile(stat, 0.025), u=quantile(stat, 0.975))## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 5.04 11.7data(hsb2, package="openintro") # Calculate diff hsb2 <- hsb2 %>% mutate(diff = math - science) # Generate 15000 bootstrap means centered at null scorediff_med_ht <- hsb2 %>% specify(response=diff) %>% hypothesize(null="point", mu=0) %>% generate(reps=15000, type="bootstrap") %>% calculate(stat="median") # Calculate observed median of differences scorediff_med_obs <- hsb2 %>% summarize(median(diff)) %>% pull() # Calculate p-value scorediff_med_ht %>% filter(stat > scorediff_med_obs) %>% summarize(p_val = (n() / 15000) * 2)## # A tibble: 1 x 1 ## p_val ## <dbl> ## 1 0.529
Chapter 3 - Inference for Difference in Two Parameters
Hypothesis testing for comparing two means:
- Data stem.cell are available in the openintro package - question of whether stem cells help with heart recovery in sheep
- Question is the impact of test vs. control, with each sheep having change measured, but only some having the stem cell therapy
- Question is the impact of test vs. control, with each sheep having change measured, but only some having the stem cell therapy
- For the hacker statistics approach, can randomly assign the sheep (multiple times) as test vs. control, and plot the ECDF (or similar) of the changes
- Can then compare how extreme our actual sample is relative to the hacker statitistics simulation
- Can then compare how extreme our actual sample is relative to the hacker statitistics simulation
- The library(infer) is built to help with problems like this
- library(infer) diff_ht_mean <- stem.cell %>%
- specify(__) %>% # y ~ x
- hypothesize(null = __) %>% # “independence” or “point”
- generate(reps = , type = ) %>% # “bootstrap”, “permute”, or “simulate”
- calculate(stat=“diff in means”) %>%
- …
- library(infer) diff_ht_mean <- stem.cell %>%
- For problems like this, the null hypothesis is “independence” and the generation type is “permute”
Bootstrap CI for difference in two means:
- Take a bootstrap sample from each of the two groups
- Calculate the bootstrap statistic of interest
- Repeat as needed to calculate a bootstrap interval
Comparing means with a t-test:
- Looking at the average hourly rate vs. citizenship from the ACS data
- t.test(hrly_rate ~ citizen, data=acs12, null=0, alternative=“two.sided”)
- t.test(hrly_rate ~ citizen, data=acs12, null=0, alternative=“two.sided”)
- Review of conditions required for the t-test
- Independence of observations (usually assumed with proper randomization and a sample size that is small relative to the population)
- Independence of observations across the samples (not paired)
- Skewed samples require larger sample sizes for the normality approximations to be valid
- Independence of observations (usually assumed with proper randomization and a sample size that is small relative to the population)
Example code includes:
data(stem.cell, package="openintro") str(stem.cell)## 'data.frame': 18 obs. of 3 variables: ## $ trmt : Factor w/ 2 levels "ctrl","esc": 1 1 1 1 1 1 1 1 1 2 ... ## $ before: num 35.2 36.5 39.8 39.8 41.8 ... ## $ after : num 29.5 29.5 36.2 38 37.5 ...# Calculate difference between before and after stem.cell <- stem.cell %>% mutate(change = after - before) # Calculate observed difference in means diff_mean <- stem.cell %>% # Group by treatment group group_by(trmt) %>% # Calculate mean change for each group summarize(mean_change = mean(change)) %>% # Extract pull() %>% # Calculate difference diff() # Generate 1000 differences in means via randomization diff_ht_mean <- stem.cell %>% # y ~ x specify(change ~ trmt) %>% # Null = no difference between means hypothesize(null = "independence") %>% # Shuffle labels 1000 times generate(reps = 1000, type = "permute") %>% # Calculate test statistic calculate(stat = "diff in means", order=rev(levels(stem.cell$trmt))) # Calculate p-value diff_ht_mean %>% # Identify simulated test statistics at least as extreme as observed filter(stat > diff_mean) %>% # Calculate p-value summarize(p_val = (n() / 1000))## # A tibble: 1 x 1 ## p_val ## <dbl> ## 1 0# Remove subjects with missing habit ncbirths_complete_habit <- ncbirths %>% filter(!is.na(habit)) # Calculate observed difference in means diff_mean <- ncbirths_complete_habit %>% # Group by habit group group_by(habit) %>% # Calculate mean weight for each group summarize(mean_weight = mean(weight)) %>% # Extract pull() %>% # Calculate difference diff() # Generate 1000 differences in means via randomization diff_ht_mean <- ncbirths_complete_habit %>% # y ~ x specify(weight ~ habit) %>% # Null = no difference between means hypothesize(null = "independence") %>% # Shuffle labels 1000 times generate(reps = 1000, type = "permute") %>% # Calculate test statistic calculate(stat = "diff in means", order=rev(levels(ncbirths_complete_habit$habit))) # Calculate p-value diff_ht_mean %>% # Identify simulated test statistics at least as extreme as observed filter(stat < diff_mean) %>% # Calculate p-value summarize(p_val = (n()/1000) * 2)## # A tibble: 1 x 1 ## p_val ## <dbl> ## 1 0.0280# Generate 1500 bootstrap difference in means diff_mean_ci <- ncbirths_complete_habit %>% specify(weight ~ habit) %>% generate(reps = 1500, type = "bootstrap") %>% calculate(stat = "diff in means", order=rev(levels(ncbirths_complete_habit$habit))) # Calculate the 95% CI via percentile method diff_mean_ci %>% summarize(l=quantile(stat, 0.025), u=quantile(stat, 0.975))## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 -0.583 -0.0530# Remove subjects with missing habit and weeks ncbirths_complete_habit_weeks <- ncbirths %>% filter(!is.na(habit) & !is.na(weeks)) # Generate 1500 bootstrap difference in medians diff_med_ci <- ncbirths_complete_habit_weeks %>% specify(weeks ~ habit) %>% generate(reps = 1500, type = "bootstrap") %>% calculate(stat="diff in medians", order=rev(levels(ncbirths_complete_habit_weeks$habit))) # Calculate the 92% CI via percentile method diff_med_ci %>% summarize(l=quantile(stat, 0.04), u=quantile(stat, 0.96))## # A tibble: 1 x 2 ## l u ## <dbl> <dbl> ## 1 -1.00 0# Create hrly_pay and filter for non-missing hrly_pay and citizen acs12_complete_hrlypay_citizen <- acs12 %>% mutate(hrly_pay = income / (hrs_work * 52)) %>% filter( !is.na(hrly_pay), !is.na(citizen) ) # Calculate percent missing new_n <- nrow(acs12_complete_hrlypay_citizen) old_n <- nrow(acs12) (perc_missing <- (old_n - new_n) / old_n)## [1] 0.5205# Calculate summary statistics acs12_complete_hrlypay_citizen %>% group_by(citizen) %>% summarize( x_bar = mean(hrly_pay), s = sd(hrly_pay), n = n() )## # A tibble: 2 x 4 ## citizen x_bar s n ## <fct> <dbl> <dbl> <int> ## 1 no 21.2 34.5 58 ## 2 yes 18.5 24.7 901# Plot the distributions ggplot(data = acs12_complete_hrlypay_citizen, mapping = aes(x = hrly_pay)) + geom_histogram(binwidth = 5) + facet_grid(. ~ citizen, labeller = labeller(citizen = c(no = "Non citizen", yes = "Citizen")))
# Construct 95% CI t.test(hrly_pay ~ citizen, data=acs12_complete_hrlypay_citizen, null=0, alternative="two.sided")## ## Welch Two Sample t-test ## ## data: hrly_pay by citizen ## t = 0.58058, df = 60.827, p-value = 0.5637 ## alternative hypothesis: true difference in means is not equal to 0 ## 95 percent confidence interval: ## -6.53483 11.88170 ## sample estimates: ## mean in group no mean in group yes ## 21.19494 18.52151
Chapter 4 - Comparing Many Means
Vocabulary score vary between social class:
- Data set includes wordsum (vocabular score) and class (lower, working, middle, upper)
ANOVA - Analysis of Variance:
- Example of runners in a marathon finsihing in different times based on many different factors
- Suppose that we are interested in a specific variable X (perhaps training time)
- Variability in finishing time due to X
- Variability in finishing time due to all factors other than X
- Variability in finishing time due to X
- The null hypothesis is that the means are the same across all of the groups, while the alternate hypothesis is that at least one mean is different
- Can assess the total variability of vocabulary scores as follows
- Variability between groups
- Variability within groups
- Variability between groups
- Running aov(x ~ y, data=z) will run ANOVA and report on
- myVar - between groups df, sumsq, and the like
- Residuals - within groups df, sumsq, and the like
- Can also calculate the percentage of variability explained
- The F-statistic is the key test statistic for this type of analysis
- myVar - between groups df, sumsq, and the like
Conditions for ANOVA:
- Independence - within groups (samples observations must be independent) and across groups (must be non-paired)
- Generally assumed to be OK with a properly stratified and randomized sample that is reasonably small relative to the population
- The between groups pairing can be handled with techniques not covered during this course
- Generally assumed to be OK with a properly stratified and randomized sample that is reasonably small relative to the population
- Approximate normality within each group
- Equal variance within each group
- Especially important when sample sizes are significantly different across groups
Post-hoc testing - determining which of the means are different:
- Can run t-tests for each group comparison, though this will epxlode the Type I error rate
- Can instead use a modified significance level for each individual test to maintain the desired overall Type I error rate
- The Bonferroni correction is common - newAlpha = tgtAlpha / K where K = k * (k-1) / 2 with k being the number of groups
- Can instead use a modified significance level for each individual test to maintain the desired overall Type I error rate
- Since there has been an assumption of constant variance, can use a consistent standard error and degrees of freedom for all the tests
Wrap-up:
- Simulation-based and CLM-based inference
- Single variables and bivariate variables
- Two levels and multiple levels
Example code includes:
gss <- readr::read_csv("./RInputFiles/gss_wordsum_class.csv")## Parsed with column specification: ## cols( ## wordsum = col_integer(), ## class = col_character() ## )str(gss)## Classes 'tbl_df', 'tbl' and 'data.frame': 795 obs. of 2 variables: ## $ wordsum: int 6 9 6 5 6 6 8 10 8 9 ... ## $ class : chr "MIDDLE" "WORKING" "WORKING" "WORKING" ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 2 ## .. ..$ wordsum: list() ## .. .. ..- attr(*, "class")= chr "collector_integer" "collector" ## .. ..$ class : list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"ggplot(gss, aes(x=wordsum)) + geom_histogram(binwidth=1) + facet_grid(class ~ .)
aov_wordsum_class <- aov(wordsum ~ class, data=gss) broom::tidy(aov_wordsum_class)## term df sumsq meansq statistic p.value ## 1 class 3 236.5644 78.854810 21.73467 1.560565e-13 ## 2 Residuals 791 2869.8003 3.628066 NA NAgss %>% group_by(class) %>% summarize(s = sd(wordsum))## # A tibble: 4 x 2 ## class s ## <chr> <dbl> ## 1 LOWER 2.24 ## 2 MIDDLE 1.89 ## 3 UPPER 2.34 ## 4 WORKING 1.87# Conduct the pairwise.t.test with p.adjust = "none" option (we'll adjust the significance level, not the p-value). The first argument is the response vector and the second argument is the grouping vector. pairwise.t.test(gss$wordsum, gss$class, p.adjust = "none") %>% broom::tidy()## group1 group2 p.value ## 1 MIDDLE LOWER 1.133345e-07 ## 2 UPPER LOWER 4.752521e-02 ## 3 WORKING LOWER 3.055619e-02 ## 5 UPPER MIDDLE 2.395734e-01 ## 6 WORKING MIDDLE 1.631637e-12 ## 9 WORKING UPPER 3.670775e-01
Introduction to Statistics with R: Correlation and Linear Regression
Chapter 1 - Introduction to Correlation Coefficients
How are correlation coefficients calculated?
- Can be calculated using the raw-score formula or the Z-score formula
- The general formula for calculating the correlation coefficient between two variables is
- r=cov(A,B) / [sA * sB]
- where cov(A,B) is the covariance between A and B, while sA and sB are the standard deviations
- r=cov(A,B) / [sA * sB]
- The covariance is defined as follows
- diff_A = A - mean(A)
- diff_B = B - mean(B)
- cov(A, B) = sum(diff_A * diff_B) / (length(A) - 1) # A and B need to be of the same length, so length(A) or length(B) will do
- diff_A = A - mean(A)
- The standard deviation is defined as the sample standard deviation, so (length(A) - 1) is in the denominator prior to the square root being taken
- sd_A = sqrt( sum(diff_A ** 2) / (length(A) - 1) )
Usefulness of correlation coefficients:
- Correlation can range between +1 (perfect positive correlation), -1 (perfect negative correlation), and 0 (no linear relationship)
- When variables are strongly correlated, knowing one variable can help you predict another variable
- Working memory capacity is strongly correlated with intelligence and IQ
Points of caution:
- Correlation does not imply causation
- The magnitude of correlation depends on many factors - sampling (full random vs. targeted population), measurement (reliable and valid), etc.
- Attenuation of correlation due to restriction of range - correlation on college graduates only may not work well
- Attenuation of correlation due to restriction of range - correlation on college graduates only may not work well
- Correlation coefficient is a sample statistic, just like the mean
Example code includes:
PE <- read.table("http://assets.datacamp.com/course/Conway/Lab_Data/Stats1.13.Lab.04.txt", header=TRUE)# Take a quick peek at both vectors (A <- c(1, 2, 3))## [1] 1 2 3(B <- c(3, 6, 7))## [1] 3 6 7# Save the differences of each vector element with the mean in a new variable diff_A <- A - mean(A) diff_B <- B - mean(B) # Do the summation of the elements of the vectors and divide by N-1 in order to acquire the covariance between the two vectors cov <- sum(diff_A*diff_B)/ (length(A)-1) # Square the differences that were found in the previous step sq_diff_A <- diff_A ** 2 sq_diff_B <- diff_B ** 2 # Take the sum of the elements, divide them by N-1 and consequently take the square root to acquire the sample standard deviations sd_A <- sqrt(sum(sq_diff_A)/(length(A)-1)) sd_B <- sqrt(sum(sq_diff_B)/(length(B)-1)) # Combine all the pieces of the puzzle correlation <- cov / (sd_A * sd_B) correlation## [1] 0.9607689# Check the validity of your result with the cor() command cor(A, B)## [1] 0.9607689# Read data from a URL into a dataframe called PE (physical endurance) - moved above to cache # PE <- read.table("http://assets.datacamp.com/course/Conway/Lab_Data/Stats1.13.Lab.04.txt", header=TRUE) # Summary statistics psych::describe(PE)## vars n mean sd median trimmed mad min max range skew ## pid 1 200 101.81 58.85 101.5 101.71 74.87 1 204 203 0.01 ## age 2 200 49.41 10.48 48.0 49.46 10.38 20 82 62 0.06 ## activeyears 3 200 10.68 4.69 11.0 10.57 4.45 0 26 26 0.30 ## endurance 4 200 26.50 10.84 27.0 26.22 10.38 3 55 52 0.22 ## kurtosis se ## pid -1.21 4.16 ## age -0.14 0.74 ## activeyears 0.46 0.33 ## endurance -0.44 0.77# Scatter plots plot(PE$age ~ PE$activeyears)
plot(PE$endurance ~ PE$activeyears)
plot(PE$endurance ~ PE$age)
# Correlation Analysis round(cor(PE[, !(names(PE) == "pid")]), 2)## age activeyears endurance ## age 1.00 0.33 -0.08 ## activeyears 0.33 1.00 0.33 ## endurance -0.08 0.33 1.00# Do some correlation tests. If the null hypothesis of no correlation can be rejected on a significance level of 5%, then the relationship between variables is significantly different from zero at the 95% confidence level cor.test(PE$age, PE$activeyears)## ## Pearson's product-moment correlation ## ## data: PE$age and PE$activeyears ## t = 4.9022, df = 198, p-value = 1.969e-06 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## 0.1993491 0.4473145 ## sample estimates: ## cor ## 0.3289909cor.test(PE$endurance, PE$activeyears)## ## Pearson's product-moment correlation ## ## data: PE$endurance and PE$activeyears ## t = 4.8613, df = 198, p-value = 2.37e-06 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## 0.1967110 0.4451154 ## sample estimates: ## cor ## 0.3265402cor.test(PE$endurance, PE$age)## ## Pearson's product-moment correlation ## ## data: PE$endurance and PE$age ## t = -1.1981, df = 198, p-value = 0.2323 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## -0.22097811 0.05454491 ## sample estimates: ## cor ## -0.08483813# The impact dataset is already loaded in rawImpactData <- " 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, control, control, control, control, control, control, control, control, control, control, control, control, control, control, control, control, control, control, control, control, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, concussed, 95, 90, 87, 84, 92, 89, 78, 97, 93, 90, 89, 97, 79, 86, 85, 85, 98, 95, 96, 92, 79, 85, 97, 89, 75, 75, 84, 93, 88, 97, 93, 96, 84, 89, 95, 95, 97, 95, 92, 95, 88, 82, 77, 72, 77, 79, 63, 82, 85, 66, 76, 79, 60, 59, 60, 76, 85, 83, 67, 84, 81, 85, 91, 74, 63, 68, 78, 74, 80, 73, 74, 70, 81, 72, 90, 74, 70, 63, 65, 69, 35.29, 31.47, 30.87, 41.87, 33.28, 40.73, 38.09, 31.65, 39.59, 30.53, 33.65, 37.51, 40.39, 32.88, 33.39, 35.13, 38.51, 29.64, 35.32, 27.36, 27.19, 32.66, 26.29, 28.92, 32.77, 32.92, 34.26, 36.08, 31.63, 28.89, 35.81, 33.61, 34.46, 39.18, 33.14, 33.03, 39.01, 35.06, 30.58, 38.45, 0.42, 0.63, 0.56, 0.66, 0.56, 0.81, 0.66, 0.79, 0.68, 0.60, 0.74, 0.51, 0.82, 0.59, 0.82, 0.63, 0.73, 0.57, 0.65, 1.00, 0.57, 0.71, 0.82, 0.61, 0.72, 0.50, 0.54, 0.65, 0.66, 0.71, 0.55, 0.79, 0.48, 0.55, 1.20, 0.73, 0.60, 0.84, 0.60, 0.42, 11, 7, 8, 7, 7, 6, 6, 10, 7, 10, 7, 7, 12, 2, 9, 10, 10, 8, 5, 11, 7, 9, 9, 9, 8, 9, 6, 10, 9, 7, 9, 7, 7, 10, 10, 11, 10, 5, 8, 11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 97, 86, 90, 85, 87, 91, 90, 94, 91, 93, 92, 89, 84, 81, 85, 87, 96, 93, 95, 93, 63, 79, 91, 85, 74, 72, 80, 59, 75, 90, 66, 85, 72, 82, 80, 59, 74, 62, 67, 66, 86, 80, 79, 70, 77, 85, 60, 72, 83, 68, 72, 79, 67, 71, 61, 72, 78, 85, 67, 80, 75, 79, 80, 72, 56, 66, 74, 69, 79, 73, 69, 61, 79, 66, 80, 70, 62, 54, 57, 63, 35.61, 37.01, 20.15, 33.26, 28.34, 33.47, 44.28, 36.14, 37.42, 25.19, 23.63, 26.32, 43.70, 32.40, 39.32, 35.62, 39.95, 35.62, 30.21, 30.37, 29.23, 44.45, 26.12, 27.98, 60.77, 31.91, 49.62, 35.68, 55.67, 25.70, 35.21, 33.01, 37.46, 53.20, 33.20, 34.59, 39.66, 35.09, 32.30, 44.49, 0.65, 0.49, 0.75, 0.19, 0.59, 0.48, 0.77, 0.90, 0.65, 0.59, 0.55, 0.56, 0.57, 0.69, 0.73, 0.48, 0.43, 0.37, 0.47, 0.50, 0.61, 0.65, 1.12, 0.65, 0.71, 0.79, 0.64, 0.70, 0.68, 0.73, 0.58, 0.97, 0.56, 0.51, 1.30, 0.70, 0.74, 1.24, 0.65, 0.98, 10, 7, 9, 8, 8, 5, 6, 10, 8, 11, 9, 9, 10, 3, 10, 12, 10, 9, 5, 11, 3, 6, 5, 5, 1, 9, 7, 11, 6, 3, 4, 3, 1, 7, 7, 4, 5, 2, 6, 5, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 26, 34, 27, 22, 26, 35, 43, 31, 39, 25, 31, 38, 14, 16, 33, 13, 27, 15, 19, 39" rawImpactNames <- c('subject', 'condition', 'vermem1', 'vismem1', 'vms1', 'rt1', 'ic1', 'sym1', 'vermem2', 'vismem2', 'vms2', 'rt2', 'ic2', 'sym2') splitImpactData <- stringr::str_split(rawImpactData, ",") impactRawMatrix <- matrix(data=splitImpactData[[1]], ncol=length(rawImpactNames)) colnames(impactRawMatrix) <- rawImpactNames rawImpactDF <- as.data.frame(impactRawMatrix, stringsAsFactors=FALSE) for (intCtr in c(1, 3:ncol(rawImpactDF))) { rawImpactDF[, intCtr] <- as.numeric(rawImpactDF[, intCtr]) } rawImpactDF$condition <- factor(stringr::str_replace_all(rawImpactDF$condition, " ", "")) impact <- rawImpactDF # Summary statistics entire dataset psych::describe(impact)## vars n mean sd median trimmed mad min max range ## subject 1 40 20.50 11.69 20.50 20.50 14.83 1.00 40.00 39.00 ## condition* 2 40 1.50 0.51 1.50 1.50 0.74 1.00 2.00 1.00 ## vermem1 3 40 89.75 6.44 91.00 90.44 6.67 75.00 98.00 23.00 ## vismem1 4 40 74.88 8.60 75.00 74.97 9.64 59.00 91.00 32.00 ## vms1 5 40 34.03 3.90 33.50 34.02 3.62 26.29 41.87 15.58 ## rt1 6 40 0.67 0.15 0.65 0.66 0.13 0.42 1.20 0.78 ## ic1 7 40 8.28 2.05 8.50 8.38 2.22 2.00 12.00 10.00 ## sym1 8 40 0.05 0.22 0.00 0.00 0.00 0.00 1.00 1.00 ## vermem2 9 40 82.00 11.02 85.00 82.97 9.64 59.00 97.00 38.00 ## vismem2 10 40 71.90 8.42 72.00 72.19 10.38 54.00 86.00 32.00 ## vms2 11 40 35.83 8.66 35.15 34.98 6.89 20.15 60.77 40.62 ## rt2 12 40 0.67 0.22 0.65 0.65 0.13 0.19 1.30 1.11 ## ic2 13 40 6.75 2.98 7.00 6.81 2.97 1.00 12.00 11.00 ## sym2 14 40 13.88 15.32 7.00 12.38 10.38 0.00 43.00 43.00 ## skew kurtosis se ## subject 0.00 -1.29 1.85 ## condition* 0.00 -2.05 0.08 ## vermem1 -0.70 -0.51 1.02 ## vismem1 -0.11 -0.96 1.36 ## vms1 0.08 -0.75 0.62 ## rt1 1.14 2.21 0.02 ## ic1 -0.57 0.36 0.32 ## sym1 3.98 14.16 0.03 ## vermem2 -0.65 -0.81 1.74 ## vismem2 -0.28 -0.87 1.33 ## vms2 0.86 0.65 1.37 ## rt2 0.93 1.29 0.03 ## ic2 -0.16 -1.06 0.47 ## sym2 0.44 -1.47 2.42# Calculate correlation coefficient entirecorr <- round(cor(impact$vismem2, impact$vermem2), 2) # Summary statistics subsets psych::describeBy(impact, impact$condition)## ## Descriptive statistics by group ## group: concussed ## vars n mean sd median trimmed mad min max range ## subject 1 20 30.50 5.92 30.50 30.50 7.41 21.00 40.00 19.00 ## condition* 2 20 1.00 0.00 1.00 1.00 0.00 1.00 1.00 0.00 ## vermem1 3 20 89.65 7.17 92.50 90.56 5.93 75.00 97.00 22.00 ## vismem1 4 20 74.75 8.03 74.00 74.25 8.15 63.00 91.00 28.00 ## vms1 5 20 33.20 3.62 33.09 33.27 3.32 26.29 39.18 12.89 ## rt1 6 20 0.66 0.17 0.63 0.64 0.13 0.42 1.20 0.78 ## ic1 7 20 8.55 1.64 9.00 8.62 1.48 5.00 11.00 6.00 ## sym1 8 20 0.05 0.22 0.00 0.00 0.00 0.00 1.00 1.00 ## vermem2 9 20 74.05 9.86 74.00 73.88 11.86 59.00 91.00 32.00 ## vismem2 10 20 69.20 8.38 69.50 69.62 10.38 54.00 80.00 26.00 ## vms2 11 20 38.27 10.01 35.15 37.32 7.73 25.70 60.77 35.07 ## rt2 12 20 0.78 0.23 0.70 0.74 0.11 0.51 1.30 0.79 ## ic2 13 20 5.00 2.53 5.00 4.88 2.97 1.00 11.00 10.00 ## sym2 14 20 27.65 9.07 27.00 27.75 11.12 13.00 43.00 30.00 ## skew kurtosis se ## subject 0.00 -1.38 1.32 ## condition* NaN NaN 0.00 ## vermem1 -0.79 -0.70 1.60 ## vismem1 0.45 -0.72 1.80 ## vms1 -0.13 -0.78 0.81 ## rt1 1.38 2.41 0.04 ## ic1 -0.39 -0.81 0.37 ## sym1 3.82 13.29 0.05 ## vermem2 0.07 -1.24 2.21 ## vismem2 -0.27 -1.26 1.87 ## vms2 0.77 -0.57 2.24 ## rt2 1.09 -0.10 0.05 ## ic2 0.39 -0.28 0.57 ## sym2 -0.11 -1.25 2.03 ## -------------------------------------------------------- ## group: control ## vars n mean sd median trimmed mad min max range skew ## subject 1 20 10.50 5.92 10.50 10.50 7.41 1.00 20.00 19.00 0.00 ## condition* 2 20 2.00 0.00 2.00 2.00 0.00 2.00 2.00 0.00 NaN ## vermem1 3 20 89.85 5.82 90.00 90.31 7.41 78.00 98.00 20.00 -0.41 ## vismem1 4 20 75.00 9.34 77.00 75.50 9.64 59.00 88.00 29.00 -0.46 ## vms1 5 20 34.86 4.09 34.39 34.85 4.92 27.36 41.87 14.51 0.09 ## rt1 6 20 0.67 0.13 0.66 0.67 0.13 0.42 1.00 0.58 0.47 ## ic1 7 20 8.00 2.41 7.50 8.12 2.22 2.00 12.00 10.00 -0.41 ## sym1 8 20 0.05 0.22 0.00 0.00 0.00 0.00 1.00 1.00 3.82 ## vermem2 9 20 89.95 4.36 90.50 90.06 5.19 81.00 97.00 16.00 -0.25 ## vismem2 10 20 74.60 7.76 74.50 75.00 8.15 60.00 86.00 26.00 -0.23 ## vms2 11 20 33.40 6.44 34.54 33.52 6.30 20.15 44.28 24.13 -0.25 ## rt2 12 20 0.57 0.16 0.56 0.57 0.13 0.19 0.90 0.71 -0.16 ## ic2 13 20 8.50 2.31 9.00 8.69 1.48 3.00 12.00 9.00 -0.73 ## sym2 14 20 0.10 0.31 0.00 0.00 0.00 0.00 1.00 1.00 2.47 ## kurtosis se ## subject -1.38 1.32 ## condition* NaN 0.00 ## vermem1 -0.87 1.30 ## vismem1 -1.27 2.09 ## vms1 -1.19 0.91 ## rt1 -0.02 0.03 ## ic1 -0.17 0.54 ## sym1 13.29 0.05 ## vermem2 -1.02 0.97 ## vismem2 -1.11 1.73 ## vms2 -0.77 1.44 ## rt2 0.06 0.04 ## ic2 -0.32 0.52 ## sym2 4.32 0.07# Create 2 subsets: control and concussed control <- subset(impact, condition == "control") concussed <- subset(impact, condition == "concussed") # Calculate correlation coefficients for each subset controlcorr <- round(cor(control$vismem2, control$vermem2), 2) concussedcorr <- round(cor(concussed$vismem2, concussed$vermem2), 2) # Display all values at the same time correlations <- cbind(entirecorr, controlcorr, concussedcorr) correlations## entirecorr controlcorr concussedcorr ## [1,] 0.45 0.37 0.35
Chapter 2 - Introduction to Linear Regression
Introduction to regression:
- Linear regression uses correlations to make predictions about one variable by knowing other variable(s)
- Simple regression has only a single predictor variable while multiple regression has many predictor variables
Regression equations and the R-squared value:
- Simple regression (one predictor) vs. multiple regression (2+ predictors)
- y = m (intercept) + b (slope) * x + e (error or residual)
- y = Bo + B1 * X1 + e (more generlized equation - allows for Bi * Xi for as many i as needed)
- y = m (intercept) + b (slope) * x + e (error or residual)
- R is the multiple correlation coefficient, which is the correlation between the predicted outcomes and the actual outcomes
- R^2 is the percentage of variance explained by the model
- R^2 is the percentage of variance explained by the model
- The regression equation would then be
- sym2=B0+B1(ic2)+e,
- where B0 is the intercept, B1 the slope and e the residual error
- sym2=B0+B1(ic2)+e,
- There are five values required for the calculation of a regression line for this model:
- The mean of sym2: mean_sym2,
- The mean of ic2: mean_ic2,
- The standard deviation of sym2: sd_sym2,
- The standard deviation of ic2: sd_ic2, and
- The correlation coefficient between sym2 and ic2: r
- The mean of sym2: mean_sym2,
- The general formula for the slope is:
- B1=r*sd(y)/sd(x)
- B1=r*sd(y)/sd(x)
- The general formula for the intercept is
- B0=mean(y)−B1*mean(x)
Multiple linear regression:
- Adding predictor variables (or better predictor variables) can increase the predictive power of the regression
Example code includes:
# Look at the dataset. Note that the variables we are interested in are on the 9th to 14th columns head(impact)## subject condition vermem1 vismem1 vms1 rt1 ic1 sym1 vermem2 vismem2 ## 1 1 control 95 88 35.29 0.42 11 0 97 86 ## 2 2 control 90 82 31.47 0.63 7 0 86 80 ## 3 3 control 87 77 30.87 0.56 8 0 90 79 ## 4 4 control 84 72 41.87 0.66 7 0 85 70 ## 5 5 control 92 77 33.28 0.56 7 1 87 77 ## 6 6 control 89 79 40.73 0.81 6 0 91 85 ## vms2 rt2 ic2 sym2 ## 1 35.61 0.65 10 0 ## 2 37.01 0.49 7 0 ## 3 20.15 0.75 9 0 ## 4 33.26 0.19 8 0 ## 5 28.34 0.59 8 1 ## 6 33.47 0.48 5 0# Create a correlation matrix for the dataset correlations <- cor(impact[, 9:14]) # Create the scatterplot matrix for the dataset corrplot::corrplot(correlations)
# Calculate the required means, standard deviations and correlation coefficient mean_sym2 <- mean(impact$sym2) mean_ic2 <- mean(impact$ic2) sd_sym2 <- sd(impact$sym2) sd_ic2 <- sd(impact$ic2) r <- cor(impact$ic2,impact$sym2) # Calculate the slope B_1 <- r * ( sd_sym2 )/( sd_ic2 ) # Calculate the intercept B_0 <- mean_sym2 - B_1 * mean_ic2 # Plot of ic2 against sym2 plot(x=impact$ic2, y=impact$sym2, main = "Scatterplot", ylab = "Symptoms", xlab = "Impulse Control") # Add the regression line abline(B_0, B_1, col = "red")
# Construct the regression model model_1 <- lm(impact$sym2 ~ impact$ic2) # Look at the results of the regression by using the summary function summary(model_1)## ## Call: ## lm(formula = impact$sym2 ~ impact$ic2) ## ## Residuals: ## Min 1Q Median 3Q Max ## -22.441 -8.983 -5.309 9.127 29.696 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 29.2945 5.5090 5.318 4.9e-06 *** ## impact$ic2 -2.2844 0.7483 -3.053 0.00413 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 13.91 on 38 degrees of freedom ## Multiple R-squared: 0.1969, Adjusted R-squared: 0.1758 ## F-statistic: 9.319 on 1 and 38 DF, p-value: 0.004125# Create a scatter plot of Impulse Control against Symptom Score plot(impact$sym2 ~ impact$ic2, main = "Scatterplot", ylab = "Symptoms", xlab = "Impulse Control") # Add a regression line abline(model_1, col = "red")# Multiple Regression model_2 <- lm(impact$sym2 ~ impact$ic2 + impact$vermem2) # Examine the results of the regression summary(model_2)## ## Call: ## lm(formula = impact$sym2 ~ impact$ic2 + impact$vermem2) ## ## Residuals: ## Min 1Q Median 3Q Max ## -20.274 -8.031 -2.703 6.245 27.962 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 79.7639 14.7765 5.398 4.1e-06 *** ## impact$ic2 -1.0711 0.7335 -1.460 0.152690 ## impact$vermem2 -0.7154 0.1981 -3.611 0.000898 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 12.12 on 37 degrees of freedom ## Multiple R-squared: 0.4062, Adjusted R-squared: 0.3742 ## F-statistic: 12.66 on 2 and 37 DF, p-value: 6.482e-05# Extract the predicted values predicted <- fitted(model_2) # Plotting predicted scores against observed scores plot(predicted ~ impact$sym2, main = "Scatterplot", xlab = "Observed Scores", ylab = "Predicted Scores") abline(lm(predicted ~ impact$sym2), col = "green")
Chapter 3 - Linear Regression Models (cont)
Estimation of coefficients - key concept is to minimize the residuals (specifically, residuals-squared):
- Ordinary Least Squares (OLS) is the process of minimizing the sum-squared of the residuals
- There is a total sum-squared of the dependent variable, and a model sum-squared (portion that is explained by the model)
Estimation of standardized and unstandardized regression coefficients:
- For simple regression, B1 = r * (sdY / sdX)
- If X and Y have been standardized, the B1 = r
- If X and Y have been standardized, the B1 = r
- Executing a standardized linear regression in R is very similar to executing an unstandardized linear regression but involves the extra step of standardizing the variables by using the scale() function
Assumptions of linear regression:
- Normal distribution for Y
- Linear relationship between X and Y
- Homoscedasticity (constant variance)
- Reliable/valid/representative measures for both X and Y
- General process for assessing these is to examine the residuals
Anscombe’s quartet:
- Data set with identical correlations and identical sd(x), sd(y) for 4 highly distinct data sets
- Same regression line for Y = 3 + 0.5 * X
- Same regression line for Y = 3 + 0.5 * X
- Calculate the residuals and plot them against the x variable(s)
- If there is any trend or pattern, then the assumptions for the linear regression have been violated
Example code includes:
# Create a linear regression with `ic2` and `vismem2` as regressors model_1 <- lm(impact$sym2 ~ impact$ic2 + impact$vismem2) # Extract the predicted values predicted_1 <- fitted(model_1) # Calculate the squared deviation of the predicted values from the observed values deviation_1 <- (impact$sym2 - predicted_1) ** 2 # Sum the squared deviations SSR_1 <- sum(deviation_1) SSR_1## [1] 7236.338# Create a linear regression with `ic2` and `vermem2` as regressors model_2 <- lm(impact$sym2 ~ impact$ic2 + impact$vermem2) # Extract the predicted values predicted_2 <- fitted(model_2) # Calculate the squared deviation of the predicted values from the observed values deviation_2 <- (impact$sym2 - predicted_2) ** 2 # Sum the squared deviations SSR_2 <- sum(deviation_2) SSR_2## [1] 5435.454# Create a standardized simple linear regression model_1_z <- lm(scale(impact$sym2) ~ scale(impact$ic2)) #Look at the output of this regression model summary(model_1_z)## ## Call: ## lm(formula = scale(impact$sym2) ~ scale(impact$ic2)) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1.4648 -0.5863 -0.3465 0.5958 1.9383 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -4.487e-16 1.435e-01 0.000 1.00000 ## scale(impact$ic2) -4.438e-01 1.454e-01 -3.053 0.00413 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.9078 on 38 degrees of freedom ## Multiple R-squared: 0.1969, Adjusted R-squared: 0.1758 ## F-statistic: 9.319 on 1 and 38 DF, p-value: 0.004125# Extract the R-Squared value for this regression r_square_1 <- summary(model_1_z)$r.square #Calculate the correlation coefficient corr_coef_1 <- sqrt(r_square_1) # Create a standardized multiple linear regression model_2_z <- lm(scale(impact$sym2) ~ scale(impact$ic2) + scale(impact$vismem2)) # Look at the output of this regression model summary(model_2_z)## ## Call: ## lm(formula = scale(impact$sym2) ~ scale(impact$ic2) + scale(impact$vismem2)) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1.4349 -0.5949 -0.3174 0.5331 1.9646 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -5.450e-16 1.443e-01 0.000 1.0000 ## scale(impact$ic2) -4.101e-01 1.526e-01 -2.688 0.0107 * ## scale(impact$vismem2) -1.171e-01 1.526e-01 -0.767 0.4479 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.9128 on 37 degrees of freedom ## Multiple R-squared: 0.2095, Adjusted R-squared: 0.1668 ## F-statistic: 4.904 on 2 and 37 DF, p-value: 0.01291# Extract the R-Squared value for this regression r_square_2 <- summary(model_2_z)$r.squared # Calculate the correlation coefficient corr_coef_2 <- sqrt(r_square_2) # Extract the residuals from the model residual <- resid(model_2) # Draw a histogram of the residuals hist(residual)
# Extract the predicted symptom scores from the model predicted <- fitted(model_2) # Plot the residuals against the predicted symptom scores plot(residual ~ predicted, main = "Scatterplot", xlab="Model 2 Predicted Scores", ylab="Model 2 Residuals" ) abline(lm(residual ~ predicted), col="red")
Inference for Linear Regression
Chapter 1 - Inferential Ideas
Variability in regression lines:
- Different samples would produce different regression lines; question is the magnitude of the impact of sampling variability
- Can take many bootstrap samples, calculate the regressions for each, and use broom::tidy() to get the various slopes
Research question - linear modeling for relationships between fat, carbohydrates, and calories in Starbucks food:
- Can look at either a one-sided or two-sided test of relationships among any two of the variables
- The standard error gives a sense for the uncertainty of the least-squares point estimate
- The p-value reported in the lm defaults to being a two-sided t-test for the intercept and coefficient estimates
- Can divide the p-value by 2 for a 1-sided test (which is only appropriate if the original research question was one-sided)
Variability of coefficients:
- Using the BikeTrail data from masiacData, can look at bicycle volume as a function of high temperature
- There is frequently variability in the intercept/coefficient estimates based on differences between the sample and the population
- The “tighter” the data, generally the tighter the standard errors for the intercept/coefficient estimates
- When there is less variability along the x-axis, generally the uncertainty in the slope increases
- The “tighter” the data, generally the tighter the standard errors for the intercept/coefficient estimates
Example code includes:
# Load the mosaicData package and the RailTrail data library(mosaicData) data(RailTrail) # Fit a linear model ride_lm <- lm(volume ~ hightemp, data=RailTrail) # View the summary of your model summary(ride_lm)## ## Call: ## lm(formula = volume ~ hightemp, data = RailTrail) ## ## Residuals: ## Min 1Q Median 3Q Max ## -254.562 -57.800 8.737 57.352 314.035 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -17.079 59.395 -0.288 0.774 ## hightemp 5.702 0.848 6.724 1.71e-09 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 104.2 on 88 degrees of freedom ## Multiple R-squared: 0.3394, Adjusted R-squared: 0.3319 ## F-statistic: 45.21 on 1 and 88 DF, p-value: 1.705e-09# Print the tidy model output ride_lm %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) -17.079281 59.3953040 -0.2875527 7.743652e-01 ## 2 hightemp 5.701878 0.8480074 6.7238541 1.705138e-09expData1 <- c(-4.3, 0.19, -2.59, -0.43, 0.59, -2.74, 3.09, 3.51, 0.56, 5.89, 0.36, -0.01, 2.59, 1.51, 2.89, -8.26, -0.46, 3.28, 4.85, 1.16, 3.03, 2.24, 1.78, -0.26, 4.29, 6.92, -6.34, 0.49, 3.4, 3.08, 2.1, -1.93, 3.72, 0.52, -4.65, 4.24, -1.21, 5.15, -10.43, 6.46, -2.78, 0.7, 2.93, -4.84, -7.08, -3.98, 8.27, -4.51, -5.22, -2.17, 2.32, 0.37, -2.53, 3.2, -8.02, -1.82, -6.17, 1.45, -0.19, -0.91, -2.02, 1.13, 11.2, 4.43, 0.88, -0.28, -9.29, 0.18, -6.9, 0.44, -9.1, -1.21, 11.32, -3.3, 3.56, 1.28, 5.76, -2.73, -9.69, -4.43, 5.71, 1.09, -8.28, -7.12, -0.33, -4.3, 4.16, 4.83, -0.29, -3.78, 5.03, 12.3, 4.79, 0.69, -11.06, 3.73, -6.64, -0.24, 5.08, -0.48, 0.68, 4.43, 2.11, 1.8, 2.98, -4.84, -3.9, 4.1, 0.05, -7.43, -2.41, 1.14, -1.87, 11.12, 6.26, 1.29, -4.54, 5.38, 3.09, -4.59, 8.55, -4.21, -0.92, 0.79, -3.48, -6.13, 3.58, 4.54, -4.83, -13.5, 1.58, -1.03, 1.34, -1.46, 5.53, -4.23, -6.95, 6.17, -0.89, 9.95, -4.12, 0.08, 2.49, -8.42, -2.4, -6.96, 7.92, -5.04, -0.25, -0.63, 8.4, 4.18, -4.86, 0.99, -5.54, -4.23, -2.23, 2.21, -0.05, -2.67, -1.14, 3.3, -5.48, 3.86, 2.1, 4.81, -1.09, -10.97, -16.68, -8.58, 3.78, 5.94, 0.35, 0.14, -8.6, -3.44, -5.14, -6.65, -0.49, -1.99, 3.54, 4.7, -0.61, 8.69, 0.91, 0.71, 3.6, -3.1, -2.99, 5.82, 3.84, 0.82, -2.74, -6.27, -3.03, 1.29, 1.58, 1.76, 4.64, -7.24, 1.54, 0.83, -0.6, -0.29, 0.78, -8.42, 9.76, 14.35, -1.09, -13.42, -1.72, 4.49, -0.02, -0.47, 8.93, 5.27, -6.06, 12.66, 0.53, -3.08, 0.52, -0.71, -0.39, -1.11, -1.72, 8.66, -1.41, 2.77, 1.03, -6.97, 7.57, -10.75, -0.88, -2.53, 1.64, 6.48, -1.61, -1.98, -5.91, 7.25, -1.67, 4.26, -7.22, 6.03, 2.92, 4.08, 9.65, -12.34, 1.24, 3.76, 3.25, -9.13, -3.23, 0.51, -1.52, -3.44, 6.75, -0.18, -3.92, -4.14, 1.14, 6.44, -0.32, 5.91, -3.55, -8.99, -6.38, -2.64, -1.47, -3.91, 12.07, 5.55, -7.94, 10.98, -6.57, -3.43, -1.13, 9.51, 11.19, -3.21, -3.19, -7.94, 2.4) expData2 <- c(-4.59, 6.5, 3.8, -6.42, 0.78, -2.4, -2.55, -3.2, -6.3, 4.69, -0.05, 5.71, 6.5, 3.69, -4.75, 4.87, -2.42, -5.04, 3.75, 1.69, -0.19, 8.33, 2.8, -0.09, 6.24, -3.73, -2.64, 8.11, -4.43, 4.42, 3.46, -6.71, -5.47, 6.84, 4.94, 2.23, 0.92, 1.56, -3.52, -5.42, -1.04, -4.33, -0.63, -1.72, -5.42, -8.92, -4.8, -6.53, 3.33, 3.39, 4.08, -3.03, -5.11, 7.04, -0.93, -2.56, -1.45, 8.75, -4.01, -5.87, 3.36, 5.83, 1.13, -1.25, -0.04, 0.23, 0.95, 3.16, -7.17, 12.37, -9.98, -9.73, 1.55, -8.56, 13.58, 0.56, 6.39, 2.34, -5.11, 6.48, -1.62, -1.16, -6.37, 7.48, 3.51, 4.82, 1.73, -0.48, -0.84, 2.58, -3.24, -1.33, 4.69, -0.99, 9.78, -16.75, -2.92, 10.15, -4.64, 5.66, 0.89, 2.11, 1.66, 3.78, 3.43, -1.09, -1.43, -10.07, -0.87, 4.41, -3.55, -1.66, 8.28, 8.3, 1.03, 6.42, -0.33, -2.63, -4.12, 6.68, -1.32, 10.69, 7.11, -3.75, 1.16, 5.19, -4.41, -4.13, -3.32, -8.24, -3.19, 1.1, 5.45, 2.19, -10.27, -0.87, -1.32, -2.77, 7.39, -14.48, -2.06, -3.46, -4.21, -6.55, -1.59, -0.44, -3.11, -4.21, -8.38, 0.01, 10.58, 3.05, 3.67, -2.52, 2.05, -2, 7.04, -0.42, -12.23, -0.44, -1.66, -1.31, -0.16, 1.72, -3.25, 2.56, -0.21, -1.59, 2.35, -2.5, 0.44, 8.61, 2.83, 10.75, 1.1, -0.89, 4.89, -0.91, 1.83, 3.2, -1.16, -3.23, 0.96, 2.59, 6.36, -0.53, -4.2, -1.13, 2.37, -1.06, -3.69, -0.25, 8.21, -5.84, -5.53, -3.03, -0.79, -0.72, -0.67, 3.23, -6.51, 2.06, -0.4, 0.75, -2.39, -2.27, -3.65, -7.56, 3.24, -4.05, -4.2, -5.91, 5.24, -11.65, -4.16, -5.99, 1.22, 1.32, -3.63, -0.9, -3.52, -5.25, 8.05, 4.09, -3.22, 5.71, 0.67, -5.46, -5.24, -1.7, -6.4, 0.48, 4.49, 15.97, -1.42, 2.41, -1.75, 4.77, -4.45, 0.88, 0.24, 11.64, -0.51, 1.58, 4.18, -3.51, 2.32, -2.15, -5.42, 5.6, 4.18, -4.82, -1.41, -5.32, 0.58, 1.23, -5.35, -5.88, 0.76, -2.81, 0.59, -2.26, 4.05, 0.32, 5.97, 4.22, -1.79, 3.28, -4.16, -4.88, -1.24, -7.38, -2.67, -4.56, 2.45, 4.92, 1.84, -1.6, 4.79, -4.02, -9.2, 6.78, -8.21, -0.18, -4.02, 4.84, 2.81, -2.65, -4.72, -0.83, -4.69, 7.94, 3.53, 4.25, 5.06, 7.88, -1.08, -0.78, 3.41, -10.45, 0.16, 0.13, -0.6, 1.82, 5.68, 5.7, 4.66, -5.4, 7.12, -2.49, 1.5, 1.27, -8.26, 0.58, 0.04, 3.17, -3.23, -0.66, -3.2, 1.59, -4, -1.96, -3.48, -3.4, -3.95, 4.52, 2.5, -3.37, -14.81, -3.22, -3.57, 2.44, 0.17, 4.8, -6.15, -3.4, -4.1, -2.68, 5.86, 2.92, -0.19, -4.64, 9.4, 6.49, -5.84, -6.62, 2.86, -3.56, -4.6, -4.87, 7.32, 3.82, 8.99, -1.46, 4.98, -1.41, -5.89, -8.86, 6.87, -7.25, 2.67, 2.81, -0.22, -3.37, 6.74, 3.33, 4.72, 1.02, -3.02, -4.9, 2, 3.41, 0.5, -7.36, 6.36, 4, 2.24, -6.75, -5.62, -8.14, 1.82, 6.23, -0.18, 10.71, -0.57, 1.38, 9.5, 1.12, 3.08, 0.08, -4.75, 4.23, -2.23, -0.82, 1.84, -1.15, 4.12, -5.86, -0.16, -6.5, 4.86) expData3 <- c(-0.62, 1.5, 5.44, -1.68, -10.04, 11.49, 1.48, -1.82, 1.57, 3.06, -2.36, -7.98, 0.25, -1.77, 3.32, 1.72, -7.55, 7.24, 2.78, -4.41, -2.55, -1.3, -1.49, 2.78, -4.37, -4.41, 0.57, 0.7, -0.56, 0.17, -2.52, 0.5, 2.46, -5.55, 2.98, 0.51, -0.28, 3.97, 6.74, 0.14, 3.54, 0.38, -2.69, 1.59, 3.09, -2.73, 4.93, 7.43, 1.76, 0.77, 4.54, 3.69, 5.75, -2.68, -1.01, 6.47, 1.91, -3.48, -2.91, 3.62, -3.72, 2.09, 0.63, -6.95, -0.66, -8.25, 6.6, -3.02, 3.51, 11.77, -1.78, -1.57, 5.58, -0.44, 3.07, -2.54, -3.1, 3.77, 8.05, -2.44, -0.95, 3.73, 1.64, 7.64, 3.63, 3.39, 1.71, -6.25, -3.47, 1.6, 3.49, 0.94, 0.18, -4.29, -2.62, 14.57, -1.73, 1.79, 2.54, -2.94, -0.56, 6.87, -4.81, 6.45, 4.2, 1.65, 8.4, 7.45, 7.11, 5.56, 1.06, -8.52, -7.68, -6.63, -4.09, 0.16, -6.08, -5.78, -4.46, -1.35, 3.34, -0.51, -3.65, -3.82, 0.64, 8.2, 14.07, -0.87, 3.3, 1.7, -3.17, -0.57, -1.06, 5.74, 0.79, -5.42, -2.22, 3.72, 2.88, -5.73, 0.82, -3.04, 6.11, 7.04, 2.84, 0.29, -2.37, 4.49, -5, -4.09, 0.33, 0.34, 0.81, 2.11, -1.55, -0.75, -7.49, 6.03, 0.14, 3.58, -0.67, 7.74, 5.55, 5.44, -8.21, 8.48, 2.15, 0.04, -3.68, 6.09, 4.06, 2.85, 2.47, -1.37, 3.66, 0.63, 0.46, -1.82, -7.6, 0.05, -3.03, -7.56, 1.56, 2.44, -2.56, -9.01, -0.19, -5.88, -7.51, -5.84, 3.79, -18, 5.33, -4.15, -6.26, 0.53, 15.21, 4.85, 1.98, -1.25, -1.12, -5.65, -0.96, 11.19, 2.76, -2.89, 0.49, -1.83, -2.52, -1.03, -1.54, -1.22, 4.27, -2.39, 0, -3.61, 0.93, -8.6, -4.41, 5.23, -3.77, 0.99, -6.99, -1.57, 3, -0.47, -3.44, 10.14, 2.8, 1.28, -0.16, 8, 4.47, 1.46, 0.86, -3.14, 1.47, 2.22, -0.05, -1.66, 3.6, -2.25, -5.84, 5.91, 2.39, 4.85, 5.07, -2.37, 0.86, -4.37, -3.32, 2.24, -3.78, 1.35, 0.01, -1.53, -3.88, 2.32, -4.27, -1.08, 4.45, 3.55, 1.82, 11.33, 1.49, -1.67, 0.49, -3.35, 0.26, -2.57, -2.51, -13.35, 6.11, 8.47, 3.94, -4.56, -3.28, 3.92, 5.81, -3.57, -1.75, -4.77, 4, -3.46, -2.25, -0.94, -4.16, -11.13, 5.81, -3.29, 5.69, 10.75, 4.29, -0.21, -0.38, 6.03, -1.97, -4.57, 7.61, -5.07, 3.82, 1.73, 8.15, 5.79, 0.19, 1.28, 3.23, -5.88, -10.91, 9.61, -0.47, 6.15, -6.18, -0.29, 1.76, 0.34) respData1 <- c(27.8, 39.19, 39.31, 42.6, 46.38, 40.21, 44.47, 46.24, 34.38, 69.78, 47.47, 53.41, 52.07, 47.09, 49.82, 13.05, 37.2, 54.27, 42.01, 31.94, 46.56, 10.89, 34.58, 44.43, 51.34, 44.57, 23.28, 46.32, 38.84, 50.78, 34.92, 35.1, 59.31, 40.65, 26.79, 37.85, 45.41, 52.6, 19.58, 36.63, 17.9, 63.94, 51.59, 19.05, 14.15, 37.03, 66.97, 15.58, 29.71, 43.78, 47.02, 37.27, 30.03, 43.11, 36.41, 32.44, 42.13, 46.72, 39.5, 32.4, 45.52, 26.89, 72.18, 51.75, 46.9, 41.5, 22.07, 46.82, 17.87, 50.12, 18.44, 28.21, 68.83, 24.07, 49.43, 43.31, 53.94, 26.36, 7.55, 17.13, 64.75, 36.93, 8.65, 21.06, 44.15, 40.35, 27.84, 42.75, 38.86, 21.84, 52.34, 63.13, 43.23, 38.48, 25.56, 37.81, 19.7, 32.33, 51.69, 40.01, 35.01, 60.59, 47.98, 32.92, 62.64, 15.48, 28.79, 46.04, 60.79, 32.6, 55.21, 41.05, 33.99, 58.24, 50.12, 43.4, 38.2, 58.34, 40.5, 25.68, 60.69, 44.46, 25.28, 40.56, 42.48, 32.15, 52.42, 56.78, 31.09, 18.29, 53.15, 30.62, 43.09, 35.78, 56.31, 20.42, 23.26, 48.99, 26.23, 61, 41.67, 41.04, 11.61, 41.64, 50.24, 18.98, 48.7, 17.97, 38, 50.85, 63.39, 57.49, 19.51, 54.11, 18.01, 33.74, 19.89, 44.66, 23.09, 42.45, 47.84, 39.38, 26.44, 25.24, 46.74, 33.03, 35.28, 35.73, -2.21, 20.49, 54.54, 50.42, 34.82, 47.67, 13.75, 44.62, 33.73, 31.53, 42.63, 36.64, 55.48, 49.84, 41.98, 69.24, 48.39, 39.12, 40.55, 41.95, 29.31, 34.22, 32.13, 33.6, 14.66, 23.75, 31.9, 35.76, 29, 50.02, 51.85, 13, 43.69, 45.67, 39.06, 43.92, 47.04, 11.32, 66.35, 56.47, 46.27, -0.56, 58.99, 57.85, 50.48, 24.43, 62.28, 49.07, 29.16, 63.71, 35.43, 25.9, 27.7, 40.02, 36.33, 43.11, 28.98, 51.88, 45.73, 48.29, 44.55, 28.76, 60.29, 14.26, 38.09, 43.13, 47.68, 60.55, 47.78, 29.85, 26.43, 62.71, 31.78, 38.87, 28.99, 56.19, 17.08, 44.7, 51.59, 9.56, 39.35, 48.91, 35.22, 26.53, 36.73, 43.78, 50.2, 32.55, 53.92, 33.67, 32.58, 34.44, 41.82, 51.16, 21.73, 53.09, 35.07, 6.84, 30.26, 33.74, 54.12, 32.41, 57.36, 52.16, 22, 64.32, 42.23, 51.91, 44.38, 47.45, 57.47, 48.04, 28.02, 21.08, 52.87, 30.48, 52.76, 51.07, 21.07, 37.38, 27.2, 35.8, 35.36, 50.08, 61.72, 27.94, 54.76, 57.88, 50.57, 38.38, 58.27, 13.5, 26.88, 33.78, 67.2, 31.6, 43.14, 43.19, 43.39, 43.09, 30.85, 34.1, 72.42, 15.42, 66.27, 43.71, 28.42, 13.3, 46.63, 35.42, 52.04, 55.89, 49.6, 44.02, 24.67, 46.79, 37.98, 39.42, 23.08, 26.36, 30.27, 57.94, 22.79, 60.07, 36.51, 49.22, 22.53, 29.96, 62.01, 42.3, 30.05, 55.57, 51.68, 23.05, 28.9, 47.02, 63.76, 42.48, 56.21, 44.29, 28.75, 43.53, 29.18, 14.44, 67.68, 16.92, -1, 51.54, 24.68, 77.5, 72.04, 51.44, 59.89, 5.48, 65.2, 35.03, 29.17, 25.99, 65.76, 54.67, 51.44, 51.34, 25.71, 25.48, 45.49, 37.31, 25.55, 59.4, 23.38, 46.47, 5.27, 48.51, 59.98, 34.85, 48.4, 62.56, 27.1, 41.17, 60.38, 57.21, 17.7, 39.84, 9.25, 39.82, 60.86, 53.29, 33.74, 66.61, 66.06, 50.5, 67.98, 21.79, 25.02, 48.24, 69.45, 35.39, 67.24, 53.73, 25.21, 43.4, 50.39, 30.88, 44.33, 6.28) respData2 <- c(18.26, 36.11, 25.88, 46.64, 38.22, -2.89, 24.17, 32.85, 37.09, 52.34, 12.35, 50.37, 31.17, 22.07, 42.49, 51.39, 34.57, 25.83, 28.45, 37.21, 41.36, 71.02, 46.11, 39.2, 36.16, 41.46, 40.83, 59.73, 30.75, 1.55, 26.67, 49.85, 35.61, 50.58, 39.23, 40.37, 45.63, 35.29, 46.06, 44.51, 36.47, 51.52, 46.69, 36.55, 53.82, 66.62, 47.33, 54.24, 27.13, 45.48, 53.06, 21.14, 52.55, 51.62, 47.59, 40.92, 27.66, 34.75, 30.73, 61.73, 36.3, 48.03, 34.31, 61.52, 27.52, 32.32, 45.33, 31.56, 32.85, 33.09, 44.52, 15.36, 48.63, 37.8, 45.67, 42.7, 40.2, 13.56, 29.57, 48.77, 29.23, 34.8, 26.91, 50.36, 29.04, 27.91, 0.98, 44.37, 29.11, 36.25, 37.73, 36.64, 18.95, 41.73, 54.34, 34.53, 56.36, 44.06, 39.32, 21.44, 53.32, 27.81, 49.95, 49.67, 68.74, 31.31, 57.06, 33.94, 39.64, 7.46, 52.8, 34.4, 70.42, 32.35, 49.45, 47.13, 34.96, 42.26, 38.28, 28.98, 37.15, 49.47, 29.23, 31.53, 28.17, 35.08, 32.34, 33.13, 33.98, 31.66, 29.63, 38.07, 49.42, 48.03, 45.81, 42.76, 63.01, 31.79, 36.26, 35.28, 34.19, 24.96, 9.13, 30.69, 36.83, 22.96, 52.03, 52.85, 49.96, 54.53, 31.88, 14.11, 51.02, 28.36, 40.92, 53.8, 63.55, 49.42, 16.49, 26.25, 34.56, 34.24, 29.5, 56.65, 33.47, 57.91, 54.78, 40.52, 41.14, 43.87, 28.43, 25.15, 38.2, 52.35, 40.83, 58.92, 37.48, 50.09, 33.76, 46.91, 30.51, 52.1, 45.28, 25.65, 28.95, 43.69, 49.32, 32.96, 34.64, 45.21, 30.77, 43.83, 45.89, 27.21, 38.51, 23.67, 37.26, 49.04, 42.06, 7.7, 36.93, 20.52, 29.9, 30.13, 55.83, 18.76, 35.06, 36.68, 34.92, 59.64, 41.81, 22.45, 28.44, 77.59, 59.44, 19.26, 34.14, 63.37, 24.93, 24.94, 16.79, 38.96, 34.77, 55, 43.88, 43.47, 29, 38.99, 20.2, 59.86, 42.71, 52.67, 47.27, 42.85, 15.67, 54, 68.67) respData3 <- c(52.15, 49.27, 49.14, 40.6, 50.46, 58.2, 45.66, 20.46, 49.97, 42.83, 25.73, 28.75, 27.69, 22.29, 53.53, 52.18, 41.86, 69.86, 37.78, 31.75, 46.9, 32.98, 52.79, 35.54, 34.51, 26.63, 39.69, 27, 47.79, 19.5, 45.63, 26.92, 44.14, 16.93, 42.78, 56.02, 47.01, 46.89, 19.48, 25.95, 61.74, 47.83, 45.1, 41.11, 64.28, 27.7, 8.05, 49.37, 43.05, 49.23, 33.99, 25.97, 60.66, 44.42, 37.06, 40.95, 21.97, 18.88, 34.68, 41.47, 35.65, 36.49, 31.45, 36.02, 38.67, 28.87, 47.33, 48.99, 39.26, 59.34, 57.07, 39.02, 35.12, 50.94, 48.7, 59.07, 36.1, 41.1, 48.53, 36.84, 20.57, 67.63, 52.86, 28.83, 47.47, 43.59, 63.2, 59.74, 23.63, 44.15, 49.71, 41.84, 28.01, 24.4, 36.04, 46.16, 39.15, 50.76, 27.61, 11.67, 27.04, 39.95, 19.58, 31.08, 70.48, 41.18, 21.73, 51.49, 45.3, 61.05, 18.11, 26.11, 41.49, 51.15, 45.32, 35.03, 45.41, 48.5, 41.7, 56.59, 44.21, 51.48, 22.45, 38.89, 42.35, 47.44, 44.4, 47.43, 38.98, 15.87, 73.93, 57.64, 55.71, 56.87, 40.79, 35.31, 56.81, 31.64, 51.35, 40.61, 44.84, 63.09, 56.03, 36.39, 54.85, 35.81, 5.16, 25.41, 26.6, 39.91, 18.4, 30.47, 28.81, 35.49, 21.98, 54.58, 37.43, 45.14, 26.94, 42.56, 61.86, 63.86, 22.83, 41.93, 43.3, 35.72, 30.3, 23.57, 62.44, 28.81, 45.11, 38.81, 37.09, 35.89, 28.37, 25.34, 20.95, 37.1, 50.87, 44.03, 25.36, 46.32, 40.63, 13.77, 33.89, 35.41, 42.71, 57.71, 46.37, 48.02, 42.58, 20.81, 64.81, 42.34, 44.93, 24.07, 71.44, 49.04, 65.73, 30.5, 56.97, 77.54, 37.53, 49.04, 52.37, 44.21, 49.64, 41.62, 33.15, 26.12, 59.01, 50.39, 12.17, 26.18, 51.92, 35.1, 45.64, 62.54, 47.18, 23.61, 13.8, 48.7, 22.64, 23.25, 26.07, 65.36, -3.6, 44.56, 20.4, 28.77, 44.92, 52.98, 38.6, 46.51, 37.8, 41.54, 14.45, 40.69, 61.11, 55.09, 30.34, 39.57, 32.23, 48.52, 44.47, 27.24, 40.09, 48.87, 31.78, 44.45, 50.95, 36.98, 24.06, 22, 39.89, 33.79, 52.91, 18.36, 32.73, 65.36, 39.55, 36.56, 61.26, 70.61, 48.07, 41.77, 83.77, 62.77, 36.37, 38.26, 31.61, 46.1, 58.54, 24.53, 39.71, 58.49, 36, 28.21, 41.23, 31.57, 31.77, 42.54, 22.47, 48.5, 46.3, 30.97, 53.55, 23.35, 60.6, 41.07, 46.19, 22.14, 52.2, 29.76, 42.34, 43.52, 38.48, 49.56, 59.15, 29.13, 20.9, 40.13, 25.61, 56.45, 35.77, 34.6, 27.61, 37.08, 42.26, 36.76, 23.48, 27.94, 43.68, 49.03, 54.34, 57.83, 45.74, 54, 41.25, 36.96, 56.99, 25.3, 37.1, 54.32, 39.65, 59.93, 53.08, 32.52, 25.58, 34.33, 50.6, 49.97, 23.38, 66.08, 39.08, 47.11, 54.24, 55.54, 45.4, 44.52, 36.92, 45.72, 29.69, 13.78, 41.59, 27.96, 40.98, 19.86, 37.85, 23.43, 41.28) popdata <- data.frame(explanatory=c(expData1, expData2, expData3), response=c(respData1, respData2, respData3) ) str(popdata)## 'data.frame': 1000 obs. of 2 variables: ## $ explanatory: num -4.3 0.19 -2.59 -0.43 0.59 -2.74 3.09 3.51 0.56 5.89 ... ## $ response : num 27.8 39.2 39.3 42.6 46.4 ...# Plot the whole dataset ggplot(popdata, aes(x = explanatory, y = response)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
# Take 2 samples of size 50 set.seed(4747) sample1 <- popdata %>% sample_n(50) sample2 <- popdata %>% sample_n(50) # Plot sample1 plot1 <- ggplot(sample1, aes(x = explanatory, y = response)) + geom_point(color = "blue") + geom_smooth(method = "lm", se = FALSE, color = "blue") plot1
# Plot sample2 over sample1 plot1 + geom_point(data = sample2, aes(x = explanatory, y = response), color = "red") + geom_smooth(data = sample2, aes(x = explanatory, y = response), method = "lm", se = FALSE, color = "red")
# Repeatedly sample the population manysamples <- infer::rep_sample_n(popdata, size=50, reps=100) # Plot the regression lines ggplot(manysamples, aes(x=explanatory, y=response, group=replicate)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
# Fit and tidy many linear models manylms <- manysamples %>% group_by(replicate) %>% do(lm(response ~ explanatory, data=.) %>% broom::tidy()) %>% filter(term=="explanatory") # Plot a histogram of the slope coefficients ggplot(manylms, aes(x=estimate)) + geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Take 100 samples of size 50 manysamples1 <- infer::rep_sample_n(popdata, size=50, reps=100) # Plot the regression line for each sample ggplot(manysamples1, aes(x=explanatory, y=response, group=replicate)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
# Take 100 samples of size 10 manysamples2 <- infer::rep_sample_n(popdata, size=10, reps=100) # Plot the regression line for each sample ggplot(manysamples2, aes(x=explanatory, y=response, group=replicate)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
# In order to understand the sampling distribution associated with the slope coefficient, it is valuable to visualize the impact changes in the sample and population have on the slope coefficient. Here, reducing the variance associated with the response variable around the line changes the variability associated with the slope statistics. # The new popdata is already loaded in your workspace. # Take 100 samples of size 50 oldPopData <- popdata popdata$response <- (oldPopData$response - mean(oldPopData$response)) / sd(oldPopData$response) popdata$response <- 40 + popdata$response * 11.152 manysamples <- infer::rep_sample_n(popdata, size=50, reps=100) # Plot a regression line for each sample ggplot(manysamples, aes(x=explanatory, y=response, group=replicate)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
Chapter 2 - Simulation Based Inference for Slope Parameters
Simulation-based inference - using the twins study from the 1920s (one twin was raise by their parents and the other in a foster home):
- The regression analysis looks at Foster vs Biological on an IQ test
- Can instead permute the response variable (Foster) and see what baseline variability in slopes would be
- Can then assess the likelihood of seeing the actual regression slope in the data; that becomes the p-value
- Can then assess the likelihood of seeing the actual regression slope in the data; that becomes the p-value
- Can run the analyses using the “infer” package; specifically
- twins %>%
- specify(Foster ~ Biological) %>%
- hypothesize(null = “independence”) %>%
- generate(reps = 10, type = “permute”) %>%
- calculate(stat = “slope”)
- twins %>%
- Typically, to do inference, you will need to know the sampling distribution of the slope under the hypothesis that there is no relationship between the explanatory and response variables
- In most situations, you don’t know the population from which the data came, so the null sampling distribution must be derived from only the original dataset
- In this exercise you’ll use the pull() function. This function takes a data frame and returns a selected column as a vector (similar to $)
- In most situations, you don’t know the population from which the data came, so the null sampling distribution must be derived from only the original dataset
Simulation-based inference for slope - can also be calculated using bootstrap for CI (as opposed to testing a null-hypothesis):
- The bootstrap will count some sets of (x, y) 2+ times, and some sets of (x, y) 0 times; there is no permuting of the data, though
- While the permuted slopes were all centered around zero (as null hypotheses), the resamples slopes will be centered around the test-statistic slope (as confidence intervals)
- Can run the analyses using the “infer” package; specifically
- twins %>%
- specify(Foster ~ Biological) %>%
- generate(reps = 100, type = “bootstrap”) %>%
- calculate(stat = “slope”)
- twins %>%
Example code includes:
# Load the infer package library(infer) twins <- readr::read_csv("./RInputFiles/twins.csv")## Parsed with column specification: ## cols( ## Foster = col_integer(), ## Biological = col_integer(), ## Social = col_character() ## )str(twins)## Classes 'tbl_df', 'tbl' and 'data.frame': 27 obs. of 3 variables: ## $ Foster : int 82 80 88 108 116 117 132 71 75 93 ... ## $ Biological: int 82 90 91 115 115 129 131 78 79 82 ... ## $ Social : chr "high" "high" "high" "high" ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 3 ## .. ..$ Foster : list() ## .. .. ..- attr(*, "class")= chr "collector_integer" "collector" ## .. ..$ Biological: list() ## .. .. ..- attr(*, "class")= chr "collector_integer" "collector" ## .. ..$ Social : list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"# Calculate the observed slope obs_slope <- lm(Foster ~ Biological, data=twins) %>% broom::tidy() %>% filter(term == "Biological") %>% pull(estimate) # Simulate 10 slopes with a permuted dataset set.seed(4747) perm_slope <- twins %>% specify(Foster ~ Biological) %>% hypothesize(null = "independence") %>% generate(reps = 10, type = "permute") %>% calculate(stat = "slope") # Print the observed slope and the 10 permuted slopes obs_slope## [1] 0.901436perm_slope## # A tibble: 10 x 2 ## replicate stat ## <int> <dbl> ## 1 1 0.143 ## 2 2 0.0710 ## 3 3 -0.456 ## 4 4 0.0749 ## 5 5 0.297 ## 6 6 0.0673 ## 7 7 0.140 ## 8 8 0.164 ## 9 9 0.0971 ## 10 10 0.184# Make a dataframe with replicates and plot them! set.seed(4747) perm_slope <- twins %>% specify(Foster ~ Biological) %>% hypothesize(null = "independence") %>% generate(reps = 500, type = "permute") %>% calculate(stat = "slope") ggplot(perm_slope, aes(x=stat)) + geom_density()
# Calculate the mean and the standard deviation of the slopes mean(perm_slope$stat)## [1] 0.006285095sd(perm_slope$stat)## [1] 0.1963073# Calculate the absolute value of the slope abs_obs_slope <- lm(Foster ~ Biological, data=twins) %>% broom::tidy() %>% filter(term == "Biological") %>% pull(estimate) %>% abs() # Compute the p-value perm_slope %>% mutate(abs_perm_slope=abs(stat)) %>% summarize(p_value = mean(abs_perm_slope > abs_obs_slope))## # A tibble: 1 x 1 ## p_value ## <dbl> ## 1 0# Calculate 1000 bootstrapped slopes set.seed(4747) BS_slope <- twins %>% specify(Foster ~ Biological) %>% generate(reps = 1000, type = "bootstrap") %>% calculate(stat = "slope") # Look at the head of BS_slope head(BS_slope)## # A tibble: 6 x 2 ## replicate stat ## <int> <dbl> ## 1 1 0.946 ## 2 2 0.966 ## 3 3 0.870 ## 4 4 0.930 ## 5 5 0.807 ## 6 6 0.900# Create a confidence interval BS_slope %>% summarize(lower = mean(stat) - 2 *sd(stat), upper = mean(stat) + 2 *sd(stat))## # A tibble: 1 x 2 ## lower upper ## <dbl> <dbl> ## 1 0.719 1.08# Set alpha alpha <- 0.05 # Create a confidence interval BS_slope %>% summarize(low = quantile(stat, alpha/2), high = quantile(stat, 1 - alpha/2))## # A tibble: 1 x 2 ## low high ## <dbl> <dbl> ## 1 0.724 1.08
Chapter 3 - t-Based Inference for the Slope Parameter
Mathematical approximation for testing and estimating slope parameters (based on the t-distribution):
- Continuing with the Starbucks data for the relationships between fat and calories
- Can look at the histogram of 1) simulation slopes for the permuted null hypothesis, and 2) t-distribution for df=n-2; should see very good overlaps
- The dt function is the density of the t-distribution; this is used by R behind the scenese to select the p-value
- The dt function is the density of the t-distribution; this is used by R behind the scenese to select the p-value
- Sometimes, the t-distribution is not such a good match to the histograms of the null hypothesis
- R will use the t-distribution even if it is not a perfect fit; further reinforcement of the value of plotting the data
- The differences between the mathematical model and the data is only really important in the edge cases (minor, low-power impact)
- R will use the t-distribution even if it is not a perfect fit; further reinforcement of the value of plotting the data
- In thinking about the scientific research question, if IQ is caused only by genetics, then we would expect the slope of the line between the two sets of twins to be 1
- Testing the hypothesized slope value of 1 can be done by making a new test statistic which evaluates how far the observed slope is from the hypothesized value of 1
- newt=(slope−1) / SE
- If the hypothesis that the slope equals one is true, then the new test statistic will have a t-distribution which we can use for calculating a p-value
- Testing the hypothesized slope value of 1 can be done by making a new test statistic which evaluates how far the observed slope is from the hypothesized value of 1
- When technical conditions (see next chapter) hold, the inference from the randomization test and the t-distribution test should give equivalent conclusions
- They will not provide the exact same answer because they are based on different methods
- But they should give p-values and confidence intervals that are reasonably close
- They will not provide the exact same answer because they are based on different methods
Intervals in regression - estimating the coefficients by way of confidence intervals (CI):
- CI = point-estimate +/- critical_value * SE_estimate
- Can use the tidy() call to get the CI, such as tidy(lm(…), conf.int=TRUE, conf.level=1-alpha)
Different types of intervals - often of interest to know the variability in the predicted value, not just the parameter estimates:
- There is greater uncertainty in predicted values as you get further away from the “center of mass” of the x-data
- Can use the broom::augment() call to get the range of predicted values for a new set of data
- alpha <- .05
- crit_val <- qt((1-alpha/2), df = nrow(starbucks) - 2)
- newfood <- data.frame(Fat = c(0,10,20,30))
- augment(lm(Calories ~ Fat, data=starbucks), newdata = newfood) %>%
- mutate(lowMean = .fitted - crit_val.se.fit, upMean = .fitted + crit_val.se.fit)
- alpha <- .05
- Can create plots for the population estimate error using ggplot
- ggplot(predMeans, aes(x = Fat, y = Calories)) +
- geom_point() +
- stat_smooth(method = “lm”, se = FALSE) + # can alternately set se=TRUE to get a similar plot
- geom_ribbon(aes(ymin = lowMean, ymax = upMean), alpha=.2)
- ggplot(predMeans, aes(x = Fat, y = Calories)) +
- The prediction interval can also be requested to give the CI on the individual (rather than population mean) members of the population given a specific X variable
- This is a combination of both the error in the population mean predictions AND the natural variability in the population on the y-metric
- FatCal_pred <- augment(FatCal_lm) %>%
- mutate(.se.pred = sqrt(FatCal_sig^2 + .se.fit^2))
- predResp <- FatCal_pred %>%
- mutate(lowResp = .fitted - crit_val.se.pred, upResp = .fitted + crit_val.se.pred)
- ggplot(predResp, aes(x = Fat, y = Calories)) +
- geom_point() +
- stat_smooth(method = “lm”, se = FALSE) +
- geom_ribbon(aes(ymin = lowResp, ymax = upResp), alpha = .2)
- This is a combination of both the error in the population mean predictions AND the natural variability in the population on the y-metric
Example code includes:
# twins_perm <- twins %>% # specify(Foster ~ Biological) %>% # hypothesize(null="independence") %>% # generate(reps = 10, type = "permute") %>% # calculate(stat = "slope") # The randomized slopes are given in the twins_perm dataframe # Look at the head of the data # head(twins_perm) # Plot the histogram with the t distribution # twins_perm %>% # filter(term == "Biological_perm") %>% # ggplot(aes(x=statistic)) + # geom_histogram(aes(y = ..density..), bins = 50) + # stat_function(fun = dt, color = "red", args = list(df=nrow(twins)-2)) # Tidy the model lm(Foster ~ Biological, data=twins) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) 9.207599 9.29989643 0.9900754 3.316237e-01 ## 2 Biological 0.901436 0.09633286 9.3575128 1.203600e-09# Create a one-sided p-value lm(Foster ~ Biological, data=twins) %>% broom::tidy() %>% filter(term == "Biological") %>% select(p.value) %>% mutate(p_value_1side = p.value/2)## p.value p_value_1side ## 1 1.2036e-09 6.018e-10# Test the new hypothesis lm(Foster ~ Biological, data = twins) %>% broom::tidy() %>% filter(term == "Biological") %>% mutate(statistic_test1 = (estimate - 1) / std.error, p_value_test1 = 2 * pt(abs(statistic_test1), df=nrow(twins)-2, lower.tail=FALSE))## term estimate std.error statistic p.value statistic_test1 ## 1 Biological 0.901436 0.09633286 9.357513 1.2036e-09 -1.023161 ## p_value_test1 ## 1 0.3160311# Find the p-value # perm_slope %>% # mutate(abs_perm_slope = abs(stat)) %>% # summarize(p_value = mean(abs_perm_slope > abs(obs_slope))) # Set alpha alpha <- 0.05 # Find the critical value crit_val <- qt(0.975, df = nrow(twins)-2) # Tidy the model with the confidence level alpha lm(Foster ~ Biological, data=twins) %>% broom::tidy(conf.int=TRUE, conf.level=1-alpha)## term estimate std.error statistic p.value conf.low ## 1 (Intercept) 9.207599 9.29989643 0.9900754 3.316237e-01 -9.9458964 ## 2 Biological 0.901436 0.09633286 9.3575128 1.203600e-09 0.7030348 ## conf.high ## 1 28.361094 ## 2 1.099837# Find the lower and upper bounds of the confidence interval lm(Foster ~ Biological, data=twins) %>% broom::tidy() %>% mutate(lower = estimate - crit_val * std.error, upper = estimate + crit_val * std.error)## term estimate std.error statistic p.value lower ## 1 (Intercept) 9.207599 9.29989643 0.9900754 3.316237e-01 -9.9458964 ## 2 Biological 0.901436 0.09633286 9.3575128 1.203600e-09 0.7030348 ## upper ## 1 28.361094 ## 2 1.099837# Create the bootstrap confidence interval BS_slope %>% summarize(low = quantile(stat, alpha/2), high = quantile(stat, 1 - alpha/2))## # A tibble: 1 x 2 ## low high ## <dbl> <dbl> ## 1 0.724 1.08# Set alpha alpha <- 0.05 # Find the critical value crit_val <- qt(1-alpha/2, nrow(twins)-2) # Create a dataframe of new observations newtwins <- data.frame(Biological = c(80, 90, 100, 110)) # Find prediction intervals lm(Foster ~ Biological, data=twins) %>% broom::augment(newdata = newtwins) %>% mutate(lowMean = .fitted - crit_val * .se.fit, upMean = .fitted + crit_val * .se.fit)## Biological .fitted .se.fit lowMean upMean ## 1 80 81.32248 2.093789 77.01024 85.63472 ## 2 90 90.33684 1.572563 87.09809 93.57559 ## 3 100 99.35120 1.554979 96.14866 102.55374 ## 4 110 108.36556 2.054014 104.13524 112.59588# Set alpha and find the critical value alpha <- 0.05 crit_val <- qt(1-alpha/2, df=nrow(twins)-2) # Find confidence intervals for the response predMeans <- lm(Foster ~ Biological, data=twins) %>% broom::augment() %>% mutate(lowMean = .fitted - crit_val*.se.fit, upMean = .fitted + crit_val*.se.fit) # Examine the intervals head(predMeans)## Foster Biological .fitted .se.fit .resid .hat .sigma ## 1 82 82 83.12535 1.962975 -1.125351 0.06449924 7.885059 ## 2 80 90 90.33684 1.572563 -10.336839 0.04139435 7.588558 ## 3 88 91 91.23827 1.543999 -3.238275 0.03990427 7.859737 ## 4 108 115 112.87274 2.411531 -4.872739 0.09734432 7.818859 ## 5 116 115 112.87274 2.411531 3.127261 0.09734432 7.859970 ## 6 117 129 125.49284 3.571300 -8.492843 0.21349013 7.642607 ## .cooksd .std.resid lowMean upMean ## 1 0.0007811552 -0.1505319 79.08253 87.16817 ## 2 0.0402839133 -1.3659358 87.09809 93.57559 ## 3 0.0037993804 -0.4275816 88.05835 94.41820 ## 4 0.0237414774 -0.6635515 107.90610 117.83938 ## 5 0.0097788932 0.4258588 107.90610 117.83938 ## 6 0.2083390637 -1.2389779 118.13761 132.84807# Plot the data with geom_ribbon() ggplot(predMeans, aes(x=Biological, y=Foster)) + geom_point() + stat_smooth(method="lm", se=FALSE) + geom_ribbon(aes(ymin = lowMean, ymax = upMean), alpha=.2)
# Plot the data with stat_smooth() ggplot(twins, aes(x = Biological, y = Foster)) + geom_point() + stat_smooth(method="lm", se=TRUE)
# Set alpha and find the critical value alpha <- 0.05 crit_val <- qt(1-alpha/2, nrow(twins)-2) # Fit a model and use glance to find sigma twin_lm <- lm(Foster ~ Biological, data=twins) twin_gl <- broom::glance(twin_lm) # Pull sigma twin_sig <- pull(twin_gl, sigma) # Augment the model to find the prediction standard errors twin_pred <- broom::augment(twin_lm) %>% mutate(.se.pred = sqrt(twin_sig ** 2 + .se.fit ** 2)) # Create prediction intervals predResp <- twin_pred %>% mutate(lowResp = .fitted - crit_val * .se.pred, upResp = .fitted + crit_val * .se.pred) # Plot the intervals using geom_ribbon() ggplot(predResp, aes(x=Biological, y=Foster)) + geom_point() + stat_smooth(method="lm", se=FALSE) + geom_ribbon(aes(ymin = lowResp, ymax = upResp), alpha = .2) + geom_ribbon(data = predMeans, aes(ymin = lowMean, ymax = upMean), alpha = .2, fill = "red")
Chapter 4 - Technical Conditions in Linear Regression
Technical conditions for linear regression:
- Inferential regression requires that the data follow the LINE
- L - linear model
- I - independent observations
- N - points normally distributed around the line
- E - equal variability around the line at all values for X
- L - linear model
- The broom::augment() will give the variables .fitted and .resid, which can be plotted against each other
- The residual vs. fitted plot frequently makes the technical violations clearer and easier to spot
- So far, you have implemented two approaches for performing inference assessment to a linear model
- The first way is given by the standard R output (lm) and is based on the t-distribution
- The derivation of the t-distribution is based on the theory (i.e., the LINE conditions)
- The derivation of the t-distribution is based on the theory (i.e., the LINE conditions)
- The second method uses a randomization test which assumes that the observations are exchangeable under the null hypothesis
- That is, when the null hypothesis (X is independent of Y) is true, the Y values can be swapped among the X values
- The technical conditions in the randomization setting are linear relationship, independent observations, and equal variances
- However, the normality assumption is not needed
- That is, when the null hypothesis (X is independent of Y) is true, the Y values can be swapped among the X values
Effect of an outlier - can have unintended impact on the inferential conclusions:
- Removing data points should never be done just to get a “more desired” fit or conclusion
- The main reasons to exclude data are 1) known incorrect data, or 2) desire to model a subset of the data
Moving forward when model assumptions are violated:
- Removing outlier points on a whim will not make for a good model
- There is a big difference in removing points that are not of interest; it is important to document in a final document WHY these points are not of interest
- It is common to be interested only in a bounded population, and to have inferential statistics that apply only within that bounded population
- There is a big difference in removing points that are not of interest; it is important to document in a final document WHY these points are not of interest
- There is sometimes a need for a transformation of X to make the linear assumptions hold - sqrt(), log(), ** 2, etc.
- The model is still considered to be a linear model, even if the transformed terms are no longer linear
Example code includes:
# A dataset containing well behaved observations has been preloaded and is called hypdata_nice. There are two variables in the dataset which are aptly named explanatory and response. expl <- c(2.14, 3.04, 2.48, 2.91, 3.12, 2.45, 3.62, 3.7, 3.11, 4.18, 3.07, 3, 3.52, 3.3, 3.58, 1.35, 2.91, 3.66, 3.97, 3.23, 3.61, 3.45, 3.36, 2.95, 3.86, 4.38, 1.73, 3.1, 3.68, 3.62, 3.42, 2.61, 3.74, 3.1, 2.07, 3.85, 2.76, 4.03, 0.91, 4.29, 2.44, 3.14, 3.59, 2.03, 1.58, 2.2, 4.65, 2.1, 1.96, 2.57, 3.46, 3.07, 2.49, 3.64, 1.4, 2.64, 1.77, 3.29, 2.96, 2.82, 2.6, 3.23, 5.24, 3.89, 3.18, 2.94, 1.14, 3.04, 1.62, 3.09, 1.18, 2.76, 5.26, 2.34, 3.71, 3.26, 4.15, 2.45, 1.06, 2.11, 4.14, 3.22, 1.34, 1.58, 2.93, 2.14, 3.83, 3.97, 2.94, 2.24, 4.01, 5.46, 3.96, 3.14, 0.79, 3.75, 1.67, 2.95, 4.02, 2.9, 3.14, 3.89, 3.42, 3.36, 3.6, 2.03, 2.22, 3.82, 3.01, 1.51, 2.52, 3.23, 2.63, 5.22, 4.25, 3.26, 2.09, 4.08, 3.62, 2.08, 4.71, 2.16, 2.82, 3.16, 2.3, 1.77, 3.72, 3.91, 2.03, 0.3, 3.32, 2.79, 3.27, 2.71, 4.11, 2.15, 1.61, 4.23, 2.82, 4.99, 2.18, 3.02, 3.5, 1.32, 2.52, 1.61, 4.58, 1.99, 2.95, 2.87, 4.68, 3.84, 2.03, 3.2, 1.89, 2.15, 2.55, 3.44, 2.99, 2.47, 2.77, 3.66, 1.9, 3.77, 3.42, 3.96, 2.78, 0.81, -0.34, 1.28, 3.76, 4.19, 3.07, 3.03, 1.28, 2.31, 1.97, 1.67, 2.9, 2.6, 3.71, 3.94, 2.88, 4.74, 3.18, 3.14, 3.72, 2.38, 2.4, 4.16, 3.77, 3.16, 2.45, 1.75, 2.39, 3.26, 3.32, 3.35, 3.93, 1.55) resp <- c(19.04, 21.44, 19.21, 20.63, 21.66, 15.99, 26.76, 28.85, 20.9, 19.16, 20.53, 22.79, 22.55, 21.72, 26.31, 18.15, 18.3, 28.03, 24.12, 20.47, 23.03, 22.06, 21.91, 20.4, 22.89, 28.62, 16.63, 22.41, 23.45, 20.06, 25.29, 15.54, 22.88, 20.3, 18.87, 26.13, 19.63, 23.3, 12.38, 27.77, 18.67, 23.12, 19.87, 20.51, 17.92, 20.25, 29.82, 13.36, 18.36, 21.2, 23.69, 17.57, 18.19, 23.12, 15.58, 18.53, 20, 21.8, 19.32, 18.8, 20.24, 24.26, 27.59, 26.02, 20.1, 17.24, 12.87, 20.05, 16.27, 19.7, 20.37, 22.49, 24.61, 23.41, 20.51, 20.4, 24, 23.17, 19.66, 17.06, 23.15, 18.48, 16.99, 14.89, 23.4, 19.94, 20.93, 24.21, 19.86, 17.71, 22.74, 25.86, 25.75, 21.4, 16.65, 25.84, 18.49, 18.95, 26, 19.74, 19.39, 25.16, 22.94, 22.01, 26.12, 19.21, 18.62, 25.96, 19.53, 15.48, 22.79, 19.91, 21.65, 29.06, 22.07, 19.59, 21.01, 26.21, 23.75, 18.61, 26.75, 17.06, 18.28, 21.06, 17.18, 17.07, 22.46, 21.55, 14.53, 10.98, 19.34, 21.72, 23.16, 21.76, 23.11, 16.42, 19.64, 24.33, 19.44, 26.39, 22.03, 19.44, 20.14, 17.29, 21.89, 17.28, 25.25, 17.96, 20.94, 21, 27.3, 20.64, 23.03, 17.6, 13.78, 19.08, 16.24, 27.76, 21.19, 21.96, 21.25, 20.94, 20.3, 22.67, 21.8, 21.34, 23.33, 15.82, 12.92, 16.54, 23.08, 24.23, 22.24, 19.79, 15.31, 20.81, 17.52, 20.92, 14.01, 18.63, 27.18, 21.96, 22.9, 26.57, 22.39, 22.09, 24.67, 20.51, 18.77, 23.92, 19.28, 21.14, 21.12, 15.82, 18.52, 25.09, 25.27, 22.47, 26.35, 16.52) hypdata_nice <- data.frame(response=resp, explanatory=expl) str(hypdata_nice)## 'data.frame': 200 obs. of 2 variables: ## $ response : num 19 21.4 19.2 20.6 21.7 ... ## $ explanatory: num 2.14 3.04 2.48 2.91 3.12 2.45 3.62 3.7 3.11 4.18 ...# Plot the data ggplot(hypdata_nice, aes(x=explanatory, y=response)) + geom_point()
# Create and augmented model nice_lm <- lm(response ~ explanatory, data=hypdata_nice) %>% broom::augment() # Print the head of nice_lm head(nice_lm)## response explanatory .fitted .se.fit .resid .hat .sigma ## 1 19.04 2.14 18.69469 0.2035616 0.3453107 0.008481517 2.215806 ## 2 21.44 3.04 21.23226 0.1568479 0.2077393 0.005035457 2.215894 ## 3 19.21 2.48 19.65333 0.1738709 -0.4433274 0.006187790 2.215717 ## 4 20.63 2.91 20.86572 0.1564783 -0.2357226 0.005011754 2.215879 ## 5 21.66 3.12 21.45782 0.1584301 0.2021774 0.005137562 2.215896 ## 6 15.99 2.45 19.56874 0.1760204 -3.5787417 0.006341730 2.201131 ## .cooksd .std.resid ## 1 1.052794e-04 0.15689185 ## 2 2.246532e-05 0.09422272 ## 3 1.260166e-04 -0.20119317 ## 4 2.878777e-05 -0.10691363 ## 5 2.171440e-05 0.09170476 ## 6 8.418718e-03 -1.62424910# Plot the residuals ggplot(nice_lm, aes(x=.fitted, y=.resid)) + geom_point()
# A dataset containing poorly behaved observations has been preloaded and is called hypdata_poor. There are two variables in the dataset which are aptly named explanatory and response resp <- c(19.08, 21.62, 19.15, 20.57, 21.84, 15.23, 29.92, 33.73, 20.66, 13.32, 20.16, 23.68, 22.54, 21.6, 29.13, 17.46, 17.2, 32.23, 24.34, 19.71, 23.2, 21.85, 21.8, 20.19, 22.25, 32.75, 16.7, 23.01, 23.8, 17.81, 27.43, 14.22, 22.57, 19.74, 18.89, 28.52, 19.39, 22.5, 13.66, 31.09, 18.52, 24.1, 17.58, 20.55, 17.67, 20.41, 34.94, 13.12, 18.35, 21.63, 24.64, 15.61, 17.88, 23.29, 15.76, 18.09, 19.68, 21.75, 18.56, 18.12, 20.38, 25.84, 27.38, 28.25, 19.27, 15.54, 13.97, 19.5, 16.38, 18.85, 18.39, 23.33, 19.43, 24.15, 18.26, 19.53, 23.52, 24.03, 17.56, 16.99, 21.78, 16.54, 16.67, 15.28, 24.62, 20.05, 18.58, 24.52, 19.41, 17.59, 21.46, 21.5, 27.59, 21.39, 15.27, 28.11, 18.25, 18.05, 27.96, 19.3, 18.25, 26.58, 23.42, 21.95, 28.78, 19.23, 18.62, 28.23, 18.78, 15.74, 23.63, 18.82, 22.2, 31.28, 19.05, 18.21, 21.14, 28.26, 24.47, 18.63, 27.6, 16.95, 17.39, 20.82, 16.92, 17.1, 21.87, 19.48, 14.47, 12.61, 17.62, 22.25, 24.02, 22.34, 21.83, 16.26, 19.09, 23.91, 19.02, 25.52, 22.33, 18.63, 18.39, 16.83, 22.5, 17.19, 24.61, 17.96, 20.98, 21.17, 29, 18.01, 23.1, 15.21, 13.99, 19.13, 15.29, 31.68, 21.3, 22.55, 21.61, 19.24, 20.18, 22.09, 21.47, 18.84, 24.5, 14.98, 10.67, 16.3, 22.91, 23.86, 22.79, 19.12, 15.5, 21.1, 17.53, 20.27, 10.99, 18.28, 30.65, 20.17, 23.9, 27.05, 22.89, 22.47, 25.97, 20.77, 18.68, 23.3, 15.71, 20.94, 21.52, 16, 18.39, 27.17, 27.45, 22.74, 28.83, 16.55) expl <- c(2.14, 3.04, 2.48, 2.91, 3.12, 2.45, 3.62, 3.7, 3.11, 4.18, 3.07, 3, 3.52, 3.3, 3.58, 1.35, 2.91, 3.66, 3.97, 3.23, 3.61, 3.45, 3.36, 2.95, 3.86, 4.38, 1.73, 3.1, 3.68, 3.62, 3.42, 2.61, 3.74, 3.1, 2.07, 3.85, 2.76, 4.03, 0.91, 4.29, 2.44, 3.14, 3.59, 2.03, 1.58, 2.2, 4.65, 2.1, 1.96, 2.57, 3.46, 3.07, 2.49, 3.64, 1.4, 2.64, 1.77, 3.29, 2.96, 2.82, 2.6, 3.23, 5.24, 3.89, 3.18, 2.94, 1.14, 3.04, 1.62, 3.09, 1.18, 2.76, 5.26, 2.34, 3.71, 3.26, 4.15, 2.45, 1.06, 2.11, 4.14, 3.22, 1.34, 1.58, 2.93, 2.14, 3.83, 3.97, 2.94, 2.24, 4.01, 5.46, 3.96, 3.14, 0.79, 3.75, 1.67, 2.95, 4.02, 2.9, 3.14, 3.89, 3.42, 3.36, 3.6, 2.03, 2.22, 3.82, 3.01, 1.51, 2.52, 3.23, 2.63, 5.22, 4.25, 3.26, 2.09, 4.08, 3.62, 2.08, 4.71, 2.16, 2.82, 3.16, 2.3, 1.77, 3.72, 3.91, 2.03, 0.3, 3.32, 2.79, 3.27, 2.71, 4.11, 2.15, 1.61, 4.23, 2.82, 4.99, 2.18, 3.02, 3.5, 1.32, 2.52, 1.61, 4.58, 1.99, 2.95, 2.87, 4.68, 3.84, 2.03, 3.2, 1.89, 2.15, 2.55, 3.44, 2.99, 2.47, 2.77, 3.66, 1.9, 3.77, 3.42, 3.96, 2.78, 0.81, -0.34, 1.28, 3.76, 4.19, 3.07, 3.03, 1.28, 2.31, 1.97, 1.67, 2.9, 2.6, 3.71, 3.94, 2.88, 4.74, 3.18, 3.14, 3.72, 2.38, 2.4, 4.16, 3.77, 3.16, 2.45, 1.75, 2.39, 3.26, 3.32, 3.35, 3.93, 1.55) hypdata_poor <- data.frame(response=resp, explanatory=expl) str(hypdata_poor)## 'data.frame': 200 obs. of 2 variables: ## $ response : num 19.1 21.6 19.1 20.6 21.8 ... ## $ explanatory: num 2.14 3.04 2.48 2.91 3.12 2.45 3.62 3.7 3.11 4.18 ...# Plot the data ggplot(hypdata_poor, aes(x=explanatory, y=response)) + geom_point()
# Create an augmented model poor_lm <- lm(response ~ explanatory, data=hypdata_poor) %>% broom::augment() # Plot the residuals ggplot(poor_lm, aes(x=.fitted, y=.resid)) + geom_point()
# The data provided in this exercise (hypdata_out) has an extreme outlier. You will run the linear model with and without the outlying point to see how one observation can affect the estimate of the line. expl <- c(2.14, 3.04, 2.48, 2.91, 3.12, 2.45, 3.62, 3.7, 3.11, 4.18, 3.07, 3, 3.52, 3.3, 3.58, 1.35, 2.91, 3.66, 3.97, 3.23, 3.61, 3.45, 3.36, 2.95, 3.86, 4.38, 1.73, 3.1, 3.68, 3.62, 3.42, 2.61, 3.74, 3.1, 2.07, 3.85, 2.76, 4.03, 0.91, 4.29, 2.44, 3.14, 3.59, 2.03, 1.58, 2.2, 4.65, 2.1, 1.96, 2.57) resp <- c(23.06, 21.85, 14.4, 27.13, 5.31, 15.71, 10.51, 26, 20.96, 22.71, 17.17, 23.26, 44.96, 30.77, 24.49, 15.47, 2.14, 23.32, 10.12, 22.58, 4.63, 19.91, 44.7, 14.24, 30.7, 27.72, 28.72, 15.84, 3.65, 13.99, 33.68, 22.02, 6.66, 7.07, 17.56, 14.94, 28.6, 33.76, 14.16, 17.31, 29.39, 46.02, 32.34, 19.49, -5.37, 26.08, 500, 17.81, 28.03, 18.73) hypdata_out <- data.frame(response=resp, explanatory=expl) str(hypdata_out)## 'data.frame': 50 obs. of 2 variables: ## $ response : num 23.06 21.85 14.4 27.13 5.31 ... ## $ explanatory: num 2.14 3.04 2.48 2.91 3.12 2.45 3.62 3.7 3.11 4.18 ...# Plot the data and a linear model ggplot(hypdata_out, aes(x=explanatory, y=response)) + geom_point() + stat_smooth(method="lm", se=FALSE)
# Remove the outlier hypdata_noout <- hypdata_out %>% filter(explanatory < 4.6) # Plot all the data and both models ggplot(hypdata_out, aes(x=explanatory, y=response)) + geom_point() + stat_smooth(method="lm", se=FALSE) + stat_smooth(data=hypdata_noout, method="lm", se=FALSE, color="red")
# Examine the tidy model lm(response ~ explanatory, data=hypdata_out) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) -46.27359 36.60763 -1.264042 0.21231907 ## 2 explanatory 24.99987 11.55650 2.163274 0.03552965# Examine the new tidy model lm(response ~ explanatory, data=hypdata_noout) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) 14.769176 6.185344 2.3877698 0.02102301 ## 2 explanatory 1.957367 1.976048 0.9905467 0.32697934# The data frames perm_slope_out and perm_slope_noout are also in your workspace. These data frames hold the permuted slopes for each of the original datsets # Finally, the observed values are stored in the variables obs_slope_out and obs_slope_noout. # Calculate the p-value with the outlier # perm_slope_out %>% # mutate(abs_perm_slope = abs(stat)) %>% # summarize(p_value = mean(abs_perm_slope > abs(obs_slope_out))) # Calculate the p-value without the outlier # perm_slope_noout %>% # mutate(abs_perm_slope = abs(stat)) %>% # summarize(p_value = mean(abs_perm_slope > abs(obs_slope_noout))) # The dataset data_nonlin has been preloaded. expl <- c(0.19, 0.34, 0.51, 0.45, 0.3, 0.92, 0.47, 0.93, 0.55, 0.93, 0.29, 0, 0.73, 0.69, 0.76, 0.86, 0.54, 0.95, 0.88, 0.64, 0.53, 0.39, 0.5, 0.53, 0.7, 0.11, 0.62, 0.69, 0.72, 0.27, 0.05, 0.45, 0.46, 0.51, 0.74, 0.86, 0.83, 0.17, 0.59, 0.45, 0.73, 0.31, 0.67, 0.83, 0.64, 0.35, 0.48, 0.2, 0.8, 0.83, 0.92, 0.34, 0.1, 0.91, 0.54, 0.08, 0.75, 0.45, 0.73, 0.11, 0.66, 0.31, 0.35, 0.18, 0.77, 0.96, 0.54, 0.59, 0.18, 0.15, 0.8, 0.21, 0.4, 0.27, 0.85, 0.64, 0.02, 0.84, 0.9, 0.42, 0.29, 0.83, 0.56, 0.78, 0.72, 0.51, 0.17, 0.18, 0.08, 0.71, 0.21, 0.97, 0.95, 0.64, 0.18, 0.55, 0.15, 0.72, 0.33, 0.73) resp <- c(11.62, 12.81, 15.01, 15.05, 10.67, 25.18, 13.41, 25.96, 16.09, 25.48, 11.75, 10.24, 22.46, 20.02, 21.09, 23.65, 14.23, 26.45, 22.89, 18.12, 13.96, 13.25, 17.5, 15.18, 20.11, 10.79, 18.75, 18.65, 17.96, 10.99, 11.33, 14.58, 12.96, 14.1, 20.47, 22.88, 23.77, 11.89, 16.97, 13.6, 21.14, 14.87, 19.76, 23.05, 15.82, 13.64, 13.56, 11.11, 23.14, 22.78, 25.49, 13.64, 10.88, 25.64, 16.58, 9.34, 19.95, 15.15, 20.22, 9.04, 18.09, 12.59, 12.51, 13.24, 22.49, 26.83, 15.11, 18.06, 11.61, 9.84, 23.75, 10.41, 13.49, 12.02, 22.65, 16.75, 10.78, 24.02, 23.93, 11.28, 12.44, 22.55, 16.57, 21.24, 21.07, 14.67, 9.53, 12.24, 10.15, 21.68, 10.5, 27.13, 26.91, 16.33, 10.58, 14.71, 12.36, 18.83, 12.6, 20) data_nonlin <- data.frame(response=resp, explanatory=expl) str(data_nonlin)## 'data.frame': 100 obs. of 2 variables: ## $ response : num 11.6 12.8 15 15.1 10.7 ... ## $ explanatory: num 0.19 0.34 0.51 0.45 0.3 0.92 0.47 0.93 0.55 0.93 ...# Create an augmented model using the non-linear data lm_nonlin <- lm(response ~ explanatory, data=data_nonlin) %>% broom::augment() # Plot the residuals ggplot(lm_nonlin, aes(x=.fitted, y=.resid)) + geom_point() + geom_abline(slope = 0, intercept = 0)
# Create a second augmented model lm2_nonlin <- lm(response ~ explanatory + I(explanatory^2), data=data_nonlin) %>% broom::augment() # Plot the second set of residuals ggplot(lm2_nonlin, aes(x=.fitted, y=.resid)) + geom_point() + geom_abline(slope = 0, intercept = 0)
# In this next example, it appears as though the variance of the response variable increases as the explanatory variable increases # Note that the fix in this exercise has the effect of changing both the variability as well as modifying the linearity of the relationship # The dataset data_nonequalvar has been preloaded expl <- c(48.9, 78.2, 39.5, 42.9, 79.9, 57.9, 35.1, 50.7, 62.6, 63.3, 38.1, 75.8, 43.6, 48.8, 78, 44.8, 29.2, 57.7, 60.6, 63.3, 47.3, 54.3, 67.3, 54.8, 52.9, 50.6, 78, 62.2, 45.8, 40.5, 52.1, 58.7, 31.1, 61.8, 52.6, 63.7, 37.8, 42.5, 29.5, 51.3, 57.2, 70.3, 46.1, 91.2, 74.9, 47.3, 34.8, 29.3, 42, 57.9, 48.1, 35, 56.7, 71.9, 47.9, 49.9, 36.3, 53.6, 50.3, 60, 53.6, 67.1, 55.6, 44.2, 56.4, 41.5, 30.4, 84.1, 48.8, 59.6, 50.8, 59.3, 94.6, 70.5, 45.4, 44.8, 36.2, 87.6, 68.9, 55.8, 64.9, 33.2, 89.9, 37.9, 54.7, 64.6, 71.6, 65.8, 48.3, 67.5, 62.1, 63.6, 67.8, 54.2, 55.6, 65.4, 55.4, 50.2, 81.3, 57.9, 62.1, 55.3, 75.5, 65, 65.6, 53.1, 71.3, 53.1, 63.3, 45.3, 61, 54, 44.8, 66.5, 55.2, 67.8, 43.2, 46.9, 57.1, 92.2, 70.1, 49.7, 46.2, 67, 29.8, 40.8, 62.6, 60.4, 86.4, 42.2, 42.9, 69.5, 63.1, 46.2, 38.5, 43.7, 53.3, 60.3, 32.6, 72.5) resp <- c(127.15, 45.06, 15.54, 26.25, 17.78, 47.81, 14.22, 104.44, 134.4, 3.21, 138.4, 59.52, 26.54, 4.43, 65.15, 21.73, 8.6, 132.58, 84.87, 242.71, 23.36, 21.16, 29.59, 100.78, 135.44, 21.39, 90.71, 12.22, 34.61, 104.3, 102.54, 9.25, 32.13, 17.37, 22.74, 20.3, 99.88, 33.7, 26.17, 9.67, 2.23, 173.31, 46.49, 339.71, 110.22, 82.22, 4.93, 6.09, 12.88, 37.66, 59.45, 5.12, 37.84, 67.36, 30.94, 30.22, 12.6, 14.14, 106.09, 52.13, 4.72, 35.19, 7.49, 35.67, 28.08, 56.13, 66.75, 69.87, 65.66, 9.08, 89.92, 20.81, 43.22, 59.37, 21.8, 34.34, 1.65, 92.08, 36.89, 63.7, 23.8, 15.55, 79.21, 35.77, 74.66, 55.85, 58.33, 41.08, 53.43, 47.58, 46.57, 23.1, 305.41, 51.99, 39.4, 49.44, 116.64, 110, 120.17, 41.52, 60.48, 26.31, 121.42, 111.76, 33.76, 43.43, 150.36, 31.19, 30.25, 74.32, 132.18, 34.32, 20.45, 106.13, 47.9, 110.07, 66.47, 19.96, 42.72, 361.12, 281.52, 139.26, 22.22, 26.9, 7.66, 3.78, 52.8, 47.61, 81.24, 80.17, 19.48, 7.72, 43.5, 51.48, 37.18, 15.6, 36.02, 6.85, 15.42, 214.95) data_nonequalvar <- data.frame(response=resp, explanatory=expl) str(data_nonequalvar)## 'data.frame': 140 obs. of 2 variables: ## $ response : num 127.2 45.1 15.5 26.2 17.8 ... ## $ explanatory: num 48.9 78.2 39.5 42.9 79.9 57.9 35.1 50.7 62.6 63.3 ...# Create an augmented model lm_nonequalvar <- lm(response ~ explanatory, data=data_nonequalvar) %>% broom::augment() # Plot the residuals ggplot(lm_nonequalvar, aes(x=.fitted, y=.resid)) + geom_point() + geom_abline(slope = 0, intercept = 0)
# Create an augmented model using the log of the response lm2_nonequalvar <- lm(log(response) ~ explanatory, data=data_nonequalvar) %>% broom::augment() # Plot the log of the resoponse ggplot(data_nonequalvar, aes(x=explanatory, y=log(response))) + geom_point() + stat_smooth(method="lm", se=FALSE)
# Plot the second set of residuals ggplot(lm2_nonequalvar, aes(x=.fitted, y=.resid)) + geom_point() + geom_abline(slope = 0, intercept = 0)
# In this last example, it appears as though the points are not normally distributed around the regression line # Again, note that the fix in this exercise has the effect of changing both the variability as well as modifying the linearity of the relationship # The dataset data_nonnorm has been preloaded resp <- c(190.58, 187.28, 172.34, 291.5, 43.66, 315.81, 94.42, 417.19, 234.56, 343.66, 127.73, 119.66, 690.5, 416.69, 334.43, 337.93, 64.21, 386.13, 176.61, 280.16, 64.13, 167.81, 578.18, 160.62, 393.2, 147.78, 432.51, 216.36, 88.97, 83.47, 226.85, 235.78, 64.11, 89.33, 295.13, 230.52, 469.54, 241.46, 246.09, 131.25, 453.87, 527.73, 422.67, 356.65, 56.87, 272, 89.89, 138.36, 488.88, 321.64, 388.48, 287.1, 161.48, 423.7, 315.97, 47.3, 207.29, 314.94, 300.95, 26.88, 215.97, 196.3, 144.08, 428.02, 516.09, 423.95, 138.73, 408.28, 202.76, 60.61, 617.6, 75.43, 177.11, 176.66, 246.48, 131.29, 170.23, 485.36, 229.54, 11.85, 200.46, 304.64, 276.42, 277.58, 468.18, 138.25, 37.26, 279.1, 101.36, 628.17, 78.01, 391.71, 462.21, 92.89, 98.36, 97.8, 317.43, 172.27, 172.11, 281.36) expl <- c(0.19, 0.34, 0.51, 0.45, 0.3, 0.92, 0.47, 0.93, 0.55, 0.93, 0.29, 0, 0.73, 0.69, 0.76, 0.86, 0.54, 0.95, 0.88, 0.64, 0.53, 0.39, 0.5, 0.53, 0.7, 0.11, 0.62, 0.69, 0.72, 0.27, 0.05, 0.45, 0.46, 0.51, 0.74, 0.86, 0.83, 0.17, 0.59, 0.45, 0.73, 0.31, 0.67, 0.83, 0.64, 0.35, 0.48, 0.2, 0.8, 0.83, 0.92, 0.34, 0.1, 0.91, 0.54, 0.08, 0.75, 0.45, 0.73, 0.11, 0.66, 0.31, 0.35, 0.18, 0.77, 0.96, 0.54, 0.59, 0.18, 0.15, 0.8, 0.21, 0.4, 0.27, 0.85, 0.64, 0.02, 0.84, 0.9, 0.42, 0.29, 0.83, 0.56, 0.78, 0.72, 0.51, 0.17, 0.18, 0.08, 0.71, 0.21, 0.97, 0.95, 0.64, 0.18, 0.55, 0.15, 0.72, 0.33, 0.73) data_nonnorm <- data.frame(response=resp, explanatory=expl) str(data_nonnorm)## 'data.frame': 100 obs. of 2 variables: ## $ response : num 190.6 187.3 172.3 291.5 43.7 ... ## $ explanatory: num 0.19 0.34 0.51 0.45 0.3 0.92 0.47 0.93 0.55 0.93 ...# Create an augmented model of the data lm_nonnorm <- lm(response ~ explanatory, data=data_nonnorm) %>% broom::augment() # Plot the residuals ggplot(lm_nonnorm, aes(x=.fitted, y=.resid)) + geom_point() + geom_abline(slope = 0, intercept = 0)
# Create the second augmented model lm2_nonnorm <- lm(sqrt(response) ~ explanatory, data=data_nonnorm) %>% broom::augment() # Plot the square root of the response ggplot(data_nonnorm, aes(x=explanatory, y=sqrt(response))) + geom_point() + stat_smooth(method="lm", se=FALSE)
# Plot the second set of residuals ggplot(lm2_nonnorm, aes(x=.fitted, y=.resid)) + geom_point() + geom_abline(slope = 0, intercept = 0)
Chapter 5 - Building on Inference in Simple Regression
Inference on transformed variables - interpretation of the coefficients is no longer just slope for y ~ x:
- With non-linear X or non-linear Y, be sure to give the right interpretation based on the transformations that were run prior to the linear regression
Multicollinearity - process of some or more of the predictor variables being correlated:
- The most common example would be when X3 = aX1 + bX2 + c
- Interpreting the coefficients can be difficult to impossible when there is multicollinearity
Multiple linear regression:
- Due to multicollinearity, coefficients can change signs and be highly misleading
- The significance of the coefficients will also change in multiple linear regression
- In simple regression, the p-value is the likelihood of no linear relationship between X and Y
- In multiple regression, the p-value for X2 is the likelihood of no linear relationship between X2 and Y GIVEN that we already have X1
- In simple regression, the p-value is the likelihood of no linear relationship between X and Y
Summary:
- Models attempt to describe populations
- Check for the key technical assumptions of LINE
- Models can be approached either as hypothesis tests or as confidence intervals
- Can use either the mathematical models, or the permutation / bootstrap approaches
Example code includes:
LAhomes <- readr::read_csv("./RInputFiles/LAhomes.csv")## Parsed with column specification: ## cols( ## city = col_character(), ## type = col_character(), ## bed = col_integer(), ## bath = col_double(), ## garage = col_character(), ## sqft = col_integer(), ## pool = col_character(), ## spa = col_character(), ## price = col_double() ## )str(LAhomes, give.attr=FALSE)## Classes 'tbl_df', 'tbl' and 'data.frame': 1594 obs. of 9 variables: ## $ city : chr "Long Beach" "Long Beach" "Long Beach" "Long Beach" ... ## $ type : chr NA NA NA NA ... ## $ bed : int 0 0 0 0 0 1 1 1 1 1 ... ## $ bath : num 1 1 1 1 1 1 1 1 1 1 ... ## $ garage: chr NA NA NA "1" ... ## $ sqft : int 513 550 550 1030 1526 552 558 596 744 750 ... ## $ pool : chr NA NA NA NA ... ## $ spa : chr NA NA NA NA ... ## $ price : num 119000 153000 205000 300000 375000 ...restNYC <- readr::read_csv("./RInputFiles/restNYC.csv")## Parsed with column specification: ## cols( ## Case = col_integer(), ## Restaurant = col_character(), ## Price = col_integer(), ## Food = col_integer(), ## Decor = col_integer(), ## Service = col_integer(), ## East = col_integer() ## )str(restNYC, give.attr=FALSE)## Classes 'tbl_df', 'tbl' and 'data.frame': 168 obs. of 7 variables: ## $ Case : int 1 2 3 4 5 6 7 8 9 10 ... ## $ Restaurant: chr "Daniella Ristorante" "Tello's Ristorante" "Biricchino" "Bottino" ... ## $ Price : int 43 32 34 41 54 52 34 34 39 44 ... ## $ Food : int 22 20 21 20 24 22 22 20 22 21 ... ## $ Decor : int 18 19 13 20 19 22 16 18 19 17 ... ## $ Service : int 20 19 18 17 21 21 21 21 22 19 ... ## $ East : int 0 0 0 0 0 0 0 1 1 1 ...# Using tidy output, run an lm analysis on price versus sqft for the LAhomes dataset. # Run one more analysis, but this time on transformed variables: log(price) versus log(sqft). # Create a tidy model lm(price ~ sqft, data=LAhomes) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) -1661892.391 64459.91198 -25.78180 8.851557e-123 ## 2 sqft 1485.995 22.70924 65.43569 0.000000e+00# Create a tidy model using the log of both variables lm(log(price) ~ log(sqft), data=LAhomes) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) 2.702788 0.14369289 18.80948 1.972382e-71 ## 2 log(sqft) 1.441583 0.01953529 73.79375 0.000000e+00# Output the tidy model lm(log(price) ~ log(sqft) + log(bath), data=LAhomes) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) 2.51405101 0.26186485 9.6005668 2.957034e-21 ## 2 log(sqft) 1.47120722 0.03952669 37.2206050 1.194181e-218 ## 3 log(bath) -0.03904504 0.04528729 -0.8621632 3.887276e-01# Using the NYC Italian restaurants dataset (compiled by Simon Sheather in A Modern Approach to Regression with R), restNYC, # you will investigate the effect on the significance of the coefficients when there are multiple variables in the model # Recall, the p-value associated with any coefficient is the probability of the observed data given that the particular variable is independent of the response AND given that all other variables are included in the model. # Output the first model lm(Price ~ Service, data=restNYC) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) -11.977811 5.1092741 -2.344327 2.024510e-02 ## 2 Service 2.818433 0.2618399 10.763954 7.879529e-21# Output the second model lm(Price ~ Service + Food + Decor, data=restNYC) %>% broom::tidy()## term estimate std.error statistic p.value ## 1 (Intercept) -24.6408955 4.7536113 -5.1836160 6.332777e-07 ## 2 Service 0.1350457 0.3956525 0.3413239 7.332967e-01 ## 3 Food 1.5555712 0.3730821 4.1695147 4.932501e-05 ## 4 Decor 1.8473352 0.2175539 8.4913900 1.170666e-14
Multiple and Logistic Regression
Chapter 1 - Parallel Slopes
What if you have two groups?
- Example of fuel efficiency vs. displacement for cars popular in 1999 through 2008
- Measurements only taken in EITHER 1999 OR 2008, and overall improvement in fuel efficiency could be driving the observed results
- Measurements only taken in EITHER 1999 OR 2008, and overall improvement in fuel efficiency could be driving the observed results
- The parallel slopes model is an example of a model with one numeric explanatory variable, and one categorical explanatory variable
- Can model as lm(response ~ continuous + factor(categorical))
Visualizing parallel slopes models:
- Can define a binary value based on the year (such as newer = (year == 2008))
- A model like lm(hwy ~ displ + factor(newer)) will help to tease out the overall change in efficiency based on 2008 vs 1999, independent of displacements
- The lines will be parallel (same slope) but with different intercepts
- The broom::augment() returns a data frame with all the observations and their key descriptive statistics variables such as .fitted
- The lines will be parallel (same slope) but with different intercepts
Interpreting parallel slopes coefficients:
- The intercept is often a rather theoretical interpretation, since X=0 may be far outside the scope of the data (or even feasible)
- The coefficient of the factor variable is the change in intercept if that factor level is present
- Be careful about which factor level is considered to be the reference - that is captured already in the overall model intercept
- The coefficient of the factor variable is the change in intercept if that factor level is present
- The slope coefficient is the estimate given to the numerical variable, and will be constant for both groups of the factor variable
- There is only ONE slope even though there are two explanatory variables
Three ways to describe a model - Mathematical, Geometric, Syntactic:
- Mathematical includes an Equation, Residuals with normality assumption, and Coefficients
- Geometric is the plotting, especially for data that reside in 2D or 3D (2D plus a factor)
- This is the hardest of the three to scale to multiple regression, given human limits in creating and visualizing complex multi-dimensional plots
- This is the hardest of the three to scale to multiple regression, given human limits in creating and visualizing complex multi-dimensional plots
- Syntactic is the format for communicating desired models to R
Example code includes:
# In this case, we want to understand how the price of MarioKart games sold at auction varies as a function of not only the number of wheels included in the package, but also whether the item is new or used # A parallel slopes model has the form y ~ x + z, where z is a categorical explanatory variable, and x is a numerical explanatory variable # Explore the data data(marioKart, package="openintro") glimpse(marioKart)## Observations: 143 ## Variables: 12 ## $ ID <dbl> 150377422259, 260483376854, 320432342985, 280405224... ## $ duration <int> 3, 7, 3, 3, 1, 3, 1, 1, 3, 7, 1, 1, 1, 1, 7, 7, 3, ... ## $ nBids <int> 20, 13, 16, 18, 20, 19, 13, 15, 29, 8, 15, 15, 13, ... ## $ cond <fct> new, used, new, new, new, new, used, new, used, use... ## $ startPr <dbl> 0.99, 0.99, 0.99, 0.99, 0.01, 0.99, 0.01, 1.00, 0.9... ## $ shipPr <dbl> 4.00, 3.99, 3.50, 0.00, 0.00, 4.00, 0.00, 2.99, 4.0... ## $ totalPr <dbl> 51.55, 37.04, 45.50, 44.00, 71.00, 45.00, 37.02, 53... ## $ shipSp <fct> standard, firstClass, firstClass, standard, media, ... ## $ sellerRate <int> 1580, 365, 998, 7, 820, 270144, 7284, 4858, 27, 201... ## $ stockPhoto <fct> yes, yes, no, yes, yes, yes, yes, yes, yes, no, yes... ## $ wheels <int> 1, 1, 1, 1, 2, 0, 0, 2, 1, 1, 2, 2, 2, 2, 1, 0, 1, ... ## $ title <fct> ~~ Wii MARIO KART & WHEEL ~ NINTENDO Wii ~ BRAN...# fit parallel slopes (mod <- lm(totalPr ~ wheels + cond, data=marioKart))## ## Call: ## lm(formula = totalPr ~ wheels + cond, data = marioKart) ## ## Coefficients: ## (Intercept) wheels condused ## 37.6673 10.2161 0.8457# The parallel slopes model mod relating total price to the number of wheels and condition is already in your workspace. # Augment the model augmented_mod <- broom::augment(mod) glimpse(augmented_mod)## Observations: 143 ## Variables: 10 ## $ totalPr <dbl> 51.55, 37.04, 45.50, 44.00, 71.00, 45.00, 37.02, 53... ## $ wheels <int> 1, 1, 1, 1, 2, 0, 0, 2, 1, 1, 2, 2, 2, 2, 1, 0, 1, ... ## $ cond <fct> new, used, new, new, new, new, used, new, used, use... ## $ .fitted <dbl> 47.88342, 48.72916, 47.88342, 47.88342, 58.09954, 3... ## $ .se.fit <dbl> 3.532896, 2.696310, 3.532896, 3.532896, 3.375000, 5... ## $ .resid <dbl> 3.666579, -11.689162, -2.383421, -3.883421, 12.9004... ## $ .hat <dbl> 0.02093127, 0.01219196, 0.02093127, 0.02093127, 0.0... ## $ .sigma <dbl> 24.504954, 24.486658, 24.506118, 24.504709, 24.4820... ## $ .cooksd <dbl> 1.640983e-04, 9.543509e-04, 6.933998e-05, 1.840819e... ## $ .std.resid <dbl> 0.15174745, -0.48163062, -0.09864186, -0.16072186, ...# scatterplot, with color data_space <- ggplot(augmented_mod, aes(x = wheels, y = totalPr, color = cond)) + geom_point() # single call to geom_line() data_space + geom_line(aes(y = .fitted))
# The babies data set contains observations about the birthweight and other characteristics of children born in the San Francisco Bay area from 1960--1967 # We would like to build a model for birthweight as a function of the mother's age and whether this child was her first (parity == 0) # birthweight=β0+β1⋅age+β2⋅parity+ϵ data(babies, package="openintro") str(babies)## 'data.frame': 1236 obs. of 8 variables: ## $ case : int 1 2 3 4 5 6 7 8 9 10 ... ## $ bwt : int 120 113 128 123 108 136 138 132 120 143 ... ## $ gestation: int 284 282 279 NA 282 286 244 245 289 299 ... ## $ parity : int 0 0 0 0 0 0 0 0 0 0 ... ## $ age : int 27 33 28 36 23 25 33 23 25 30 ... ## $ height : int 62 64 64 69 67 62 62 65 62 66 ... ## $ weight : int 100 135 115 190 125 93 178 140 125 136 ... ## $ smoke : int 0 0 1 0 1 0 0 0 0 1 ...# build model lm(bwt ~ age + parity, data=babies)## ## Call: ## lm(formula = bwt ~ age + parity, data = babies) ## ## Coefficients: ## (Intercept) age parity ## 118.27782 0.06315 -1.65248# build model lm(bwt ~ gestation + smoke, data=babies)## ## Call: ## lm(formula = bwt ~ gestation + smoke, data = babies) ## ## Coefficients: ## (Intercept) gestation smoke ## -0.9317 0.4429 -8.0883
Chapter 2 - Evaluating and Extending Parallel Slopes
Model fit, residuals, and prediction:
- Model fits is evaluated based on the residuals (distance in y from the fitted line), specifically the mean residual-squared
- The coefficient of determination (R-squared) is defined as 1 - SSE/SST where SSE is the sum-squared error and SST is the sum-squared total (known prior to any modeling
- Since the R-squared will always improve with additional variables, the adjusted R-squared is frequently reported also
- Adj R^2 = 1 - (SSE / SST) * (n - 1) / (n - p - 1) where n is the number of data points and p is the number of predictor variables
- Adj R^2 = 1 - (SSE / SST) * (n - 1) / (n - p - 1) where n is the number of data points and p is the number of predictor variables
- Can gain the fitted values in either of two ways
- predict(myLM) will return the predicted values
- broom::augment(myLM) will return a data frame with many variables, one of which is .fitted
- Note that broom::augment() is considered to be part of the tidyverse and is preferred for this course
- predict(myLM) will return the predicted values
- Can make predictions for observations that are out of sample
- predict(myLM, newdata=myFrame) where myFrame is a data frame of 1+ observations with the same variables as were used for the model creation
- broom::augment(myLM, newdata=myFrame) will return a data frame including .fitted and .se.fit
- predict(myLM, newdata=myFrame) where myFrame is a data frame of 1+ observations with the same variables as were used for the model creation
Understanding interaction - idea that the model might have both different slopes and different intercepts:
* The interaction term is the product of two (or more) variables - for example Y ~ X1 + X2 + X1:X2
* The R syntax colon(:) means X1X2, or ther interaction between X1 and X2
Interaction terms can change the intepretation of the model, and also of the components of the model
* Including an interaction term in a model is easy—we just have to tell lm() that we want to include that new variable. An expression of the form
* lm(y ~ x + z + x:z, data = mydata)
* The use of the colon (:) here means that the interaction between x and z will be a third term in the model
* Interaction models are easy to visualize in the with ggplot2 because they have the same coefficients as if the models were fit independently to each group defined by the level of the categorical variable
* In this case, new and used MarioKarts each get their own regression line
* To see this, we can set an aesthetic (e.g. color) to the categorical variable, and then add a geom_smooth() layer to overlay the regression line for each colorSimpson’s Paradox:
- Example of moderating variable, such as percentage taking the SAT significantly driving average SAT scores
- SAT_wbin <- SAT %>%
- mutate(sat_bin = cut(sat_pct, 3))
- mod <- lm(formula = total ~ salary + sat_bin, data = SAT_wbin)
- SAT_wbin <- SAT %>%
- When Simpson’s paradox is present the overall slope may be of different sign (positive vs begative) than the slope within any of the properly cut sub-groups
Example code includes:
mario_kart <- marioKart %>% filter(totalPr <= 75) # fit parallel slopes (mod <- lm(totalPr ~ wheels + cond, data=mario_kart))## ## Call: ## lm(formula = totalPr ~ wheels + cond, data = mario_kart) ## ## Coefficients: ## (Intercept) wheels condused ## 42.370 7.233 -5.585# R^2 and adjusted R^2 summary(mod)## ## Call: ## lm(formula = totalPr ~ wheels + cond, data = mario_kart) ## ## Residuals: ## Min 1Q Median 3Q Max ## -11.0078 -3.0754 -0.8254 2.9822 14.1646 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 42.3698 1.0651 39.780 < 2e-16 *** ## wheels 7.2328 0.5419 13.347 < 2e-16 *** ## condused -5.5848 0.9245 -6.041 1.35e-08 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 4.887 on 138 degrees of freedom ## Multiple R-squared: 0.7165, Adjusted R-squared: 0.7124 ## F-statistic: 174.4 on 2 and 138 DF, p-value: < 2.2e-16# add random noise mario_kart_noisy <- mario_kart %>% mutate(noise = rnorm(n=n())) # compute new model mod2 <- lm(totalPr ~ wheels + cond + noise, data=mario_kart_noisy) # new R^2 and adjusted R^2 summary(mod2)## ## Call: ## lm(formula = totalPr ~ wheels + cond + noise, data = mario_kart_noisy) ## ## Residuals: ## Min 1Q Median 3Q Max ## -11.9844 -2.9707 -0.9883 2.8480 14.2079 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 42.5665 1.0701 39.777 < 2e-16 *** ## wheels 7.2002 0.5404 13.324 < 2e-16 *** ## condused -5.7546 0.9288 -6.196 6.36e-09 *** ## noise -0.5537 0.3888 -1.424 0.157 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 4.869 on 137 degrees of freedom ## Multiple R-squared: 0.7207, Adjusted R-squared: 0.7145 ## F-statistic: 117.8 on 3 and 137 DF, p-value: < 2.2e-16# return a vector predict(mod)## 1 2 3 4 5 6 7 8 ## 49.60260 44.01777 49.60260 49.60260 56.83544 42.36976 36.78493 56.83544 ## 9 10 11 12 13 14 15 16 ## 44.01777 44.01777 56.83544 56.83544 56.83544 56.83544 44.01777 36.78493 ## 17 18 19 20 21 22 23 24 ## 49.60260 49.60260 56.83544 36.78493 56.83544 56.83544 56.83544 44.01777 ## 25 26 27 28 29 30 31 32 ## 56.83544 36.78493 36.78493 36.78493 49.60260 36.78493 36.78493 44.01777 ## 33 34 35 36 37 38 39 40 ## 51.25061 44.01777 44.01777 36.78493 44.01777 56.83544 56.83544 49.60260 ## 41 42 43 44 45 46 47 48 ## 44.01777 51.25061 56.83544 56.83544 44.01777 56.83544 36.78493 36.78493 ## 49 50 51 52 53 54 55 56 ## 44.01777 56.83544 36.78493 44.01777 42.36976 36.78493 36.78493 44.01777 ## 57 58 59 60 61 62 63 64 ## 44.01777 36.78493 36.78493 56.83544 36.78493 56.83544 36.78493 51.25061 ## 65 66 67 68 69 70 71 72 ## 56.83544 44.01777 58.48345 51.25061 49.60260 44.01777 49.60260 56.83544 ## 73 74 75 76 77 78 79 80 ## 56.83544 51.25061 44.01777 36.78493 36.78493 36.78493 44.01777 56.83544 ## 81 82 83 84 85 86 87 88 ## 44.01777 65.71629 44.01777 56.83544 36.78493 49.60260 49.60260 36.78493 ## 89 90 91 92 93 94 95 96 ## 44.01777 36.78493 51.25061 44.01777 36.78493 51.25061 42.36976 56.83544 ## 97 98 99 100 101 102 103 104 ## 51.25061 44.01777 51.25061 56.83544 56.83544 56.83544 36.78493 49.60260 ## 105 106 107 108 109 110 111 112 ## 51.25061 44.01777 56.83544 49.60260 36.78493 44.01777 51.25061 56.83544 ## 113 114 115 116 117 118 119 120 ## 64.06828 44.01777 49.60260 44.01777 49.60260 51.25061 42.36976 44.01777 ## 121 122 123 124 125 126 127 128 ## 56.83544 44.01777 49.60260 44.01777 51.25061 56.83544 56.83544 49.60260 ## 129 130 131 132 133 134 135 136 ## 56.83544 36.78493 44.01777 44.01777 36.78493 56.83544 36.78493 44.01777 ## 137 138 139 140 141 ## 36.78493 51.25061 49.60260 36.78493 56.83544# return a data frame broom::augment(mod)## totalPr wheels cond .fitted .se.fit .resid .hat ## 1 51.55 1 new 49.60260 0.7087865 1.94739955 0.02103158 ## 2 37.04 1 used 44.01777 0.5465195 -6.97776738 0.01250410 ## 3 45.50 1 new 49.60260 0.7087865 -4.10260045 0.02103158 ## 4 44.00 1 new 49.60260 0.7087865 -5.60260045 0.02103158 ## 5 71.00 2 new 56.83544 0.6764502 14.16455915 0.01915635 ## 6 45.00 0 new 42.36976 1.0651119 2.63023994 0.04749321 ## 7 37.02 0 used 36.78493 0.7065565 0.23507301 0.02089945 ## 8 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 9 47.00 1 used 44.01777 0.5465195 2.98223262 0.01250410 ## 10 50.00 1 used 44.01777 0.5465195 5.98223262 0.01250410 ## 11 54.99 2 new 56.83544 0.6764502 -1.84544085 0.01915635 ## 12 56.01 2 new 56.83544 0.6764502 -0.82544085 0.01915635 ## 13 48.00 2 new 56.83544 0.6764502 -8.83544085 0.01915635 ## 14 56.00 2 new 56.83544 0.6764502 -0.83544085 0.01915635 ## 15 43.33 1 used 44.01777 0.5465195 -0.68776738 0.01250410 ## 16 46.00 0 used 36.78493 0.7065565 9.21507301 0.02089945 ## 17 46.71 1 new 49.60260 0.7087865 -2.89260045 0.02103158 ## 18 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 19 55.99 2 new 56.83544 0.6764502 -0.84544085 0.01915635 ## 20 31.00 0 used 36.78493 0.7065565 -5.78492699 0.02089945 ## 21 53.98 2 new 56.83544 0.6764502 -2.85544085 0.01915635 ## 22 64.95 2 new 56.83544 0.6764502 8.11455915 0.01915635 ## 23 50.50 2 new 56.83544 0.6764502 -6.33544085 0.01915635 ## 24 46.50 1 used 44.01777 0.5465195 2.48223262 0.01250410 ## 25 55.00 2 new 56.83544 0.6764502 -1.83544085 0.01915635 ## 26 34.50 0 used 36.78493 0.7065565 -2.28492699 0.02089945 ## 27 36.00 0 used 36.78493 0.7065565 -0.78492699 0.02089945 ## 28 40.00 0 used 36.78493 0.7065565 3.21507301 0.02089945 ## 29 47.00 1 new 49.60260 0.7087865 -2.60260045 0.02103158 ## 30 43.00 0 used 36.78493 0.7065565 6.21507301 0.02089945 ## 31 31.00 0 used 36.78493 0.7065565 -5.78492699 0.02089945 ## 32 41.99 1 used 44.01777 0.5465195 -2.02776738 0.01250410 ## 33 49.49 2 used 51.25061 0.8279109 -1.76060777 0.02869514 ## 34 41.00 1 used 44.01777 0.5465195 -3.01776738 0.01250410 ## 35 44.78 1 used 44.01777 0.5465195 0.76223262 0.01250410 ## 36 47.00 0 used 36.78493 0.7065565 10.21507301 0.02089945 ## 37 44.00 1 used 44.01777 0.5465195 -0.01776738 0.01250410 ## 38 63.99 2 new 56.83544 0.6764502 7.15455915 0.01915635 ## 39 53.76 2 new 56.83544 0.6764502 -3.07544085 0.01915635 ## 40 46.03 1 new 49.60260 0.7087865 -3.57260045 0.02103158 ## 41 42.25 1 used 44.01777 0.5465195 -1.76776738 0.01250410 ## 42 46.00 2 used 51.25061 0.8279109 -5.25060777 0.02869514 ## 43 51.99 2 new 56.83544 0.6764502 -4.84544085 0.01915635 ## 44 55.99 2 new 56.83544 0.6764502 -0.84544085 0.01915635 ## 45 41.99 1 used 44.01777 0.5465195 -2.02776738 0.01250410 ## 46 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 47 39.00 0 used 36.78493 0.7065565 2.21507301 0.02089945 ## 48 38.06 0 used 36.78493 0.7065565 1.27507301 0.02089945 ## 49 46.00 1 used 44.01777 0.5465195 1.98223262 0.01250410 ## 50 59.88 2 new 56.83544 0.6764502 3.04455915 0.01915635 ## 51 28.98 0 used 36.78493 0.7065565 -7.80492699 0.02089945 ## 52 36.00 1 used 44.01777 0.5465195 -8.01776738 0.01250410 ## 53 51.99 0 new 42.36976 1.0651119 9.62023994 0.04749321 ## 54 43.95 0 used 36.78493 0.7065565 7.16507301 0.02089945 ## 55 32.00 0 used 36.78493 0.7065565 -4.78492699 0.02089945 ## 56 40.06 1 used 44.01777 0.5465195 -3.95776738 0.01250410 ## 57 48.00 1 used 44.01777 0.5465195 3.98223262 0.01250410 ## 58 36.00 0 used 36.78493 0.7065565 -0.78492699 0.02089945 ## 59 31.00 0 used 36.78493 0.7065565 -5.78492699 0.02089945 ## 60 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 61 30.00 0 used 36.78493 0.7065565 -6.78492699 0.02089945 ## 62 58.00 2 new 56.83544 0.6764502 1.16455915 0.01915635 ## 63 38.10 0 used 36.78493 0.7065565 1.31507301 0.02089945 ## 64 61.76 2 used 51.25061 0.8279109 10.50939223 0.02869514 ## 65 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 66 40.00 1 used 44.01777 0.5465195 -4.01776738 0.01250410 ## 67 64.50 3 used 58.48345 1.2882085 6.01655183 0.06947257 ## 68 49.01 2 used 51.25061 0.8279109 -2.24060777 0.02869514 ## 69 47.00 1 new 49.60260 0.7087865 -2.60260045 0.02103158 ## 70 40.10 1 used 44.01777 0.5465195 -3.91776738 0.01250410 ## 71 41.50 1 new 49.60260 0.7087865 -8.10260045 0.02103158 ## 72 56.00 2 new 56.83544 0.6764502 -0.83544085 0.01915635 ## 73 64.95 2 new 56.83544 0.6764502 8.11455915 0.01915635 ## 74 49.00 2 used 51.25061 0.8279109 -2.25060777 0.02869514 ## 75 48.00 1 used 44.01777 0.5465195 3.98223262 0.01250410 ## 76 38.00 0 used 36.78493 0.7065565 1.21507301 0.02089945 ## 77 45.00 0 used 36.78493 0.7065565 8.21507301 0.02089945 ## 78 41.95 0 used 36.78493 0.7065565 5.16507301 0.02089945 ## 79 43.36 1 used 44.01777 0.5465195 -0.65776738 0.01250410 ## 80 54.99 2 new 56.83544 0.6764502 -1.84544085 0.01915635 ## 81 45.21 1 used 44.01777 0.5465195 1.19223262 0.01250410 ## 82 65.02 4 used 65.71629 1.7946635 -0.69628856 0.13483640 ## 83 45.75 1 used 44.01777 0.5465195 1.73223262 0.01250410 ## 84 64.00 2 new 56.83544 0.6764502 7.16455915 0.01915635 ## 85 36.00 0 used 36.78493 0.7065565 -0.78492699 0.02089945 ## 86 54.70 1 new 49.60260 0.7087865 5.09739955 0.02103158 ## 87 49.91 1 new 49.60260 0.7087865 0.30739955 0.02103158 ## 88 47.00 0 used 36.78493 0.7065565 10.21507301 0.02089945 ## 89 43.00 1 used 44.01777 0.5465195 -1.01776738 0.01250410 ## 90 35.99 0 used 36.78493 0.7065565 -0.79492699 0.02089945 ## 91 54.49 2 used 51.25061 0.8279109 3.23939223 0.02869514 ## 92 46.00 1 used 44.01777 0.5465195 1.98223262 0.01250410 ## 93 31.06 0 used 36.78493 0.7065565 -5.72492699 0.02089945 ## 94 55.60 2 used 51.25061 0.8279109 4.34939223 0.02869514 ## 95 40.10 0 new 42.36976 1.0651119 -2.26976006 0.04749321 ## 96 52.59 2 new 56.83544 0.6764502 -4.24544085 0.01915635 ## 97 44.00 2 used 51.25061 0.8279109 -7.25060777 0.02869514 ## 98 38.26 1 used 44.01777 0.5465195 -5.75776738 0.01250410 ## 99 51.00 2 used 51.25061 0.8279109 -0.25060777 0.02869514 ## 100 48.99 2 new 56.83544 0.6764502 -7.84544085 0.01915635 ## 101 66.44 2 new 56.83544 0.6764502 9.60455915 0.01915635 ## 102 63.50 2 new 56.83544 0.6764502 6.66455915 0.01915635 ## 103 42.00 0 used 36.78493 0.7065565 5.21507301 0.02089945 ## 104 47.00 1 new 49.60260 0.7087865 -2.60260045 0.02103158 ## 105 55.00 2 used 51.25061 0.8279109 3.74939223 0.02869514 ## 106 33.01 1 used 44.01777 0.5465195 -11.00776738 0.01250410 ## 107 53.76 2 new 56.83544 0.6764502 -3.07544085 0.01915635 ## 108 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 109 43.00 0 used 36.78493 0.7065565 6.21507301 0.02089945 ## 110 42.55 1 used 44.01777 0.5465195 -1.46776738 0.01250410 ## 111 52.50 2 used 51.25061 0.8279109 1.24939223 0.02869514 ## 112 57.50 2 new 56.83544 0.6764502 0.66455915 0.01915635 ## 113 75.00 3 new 64.06828 1.0000415 10.93171876 0.04186751 ## 114 48.92 1 used 44.01777 0.5465195 4.90223262 0.01250410 ## 115 45.99 1 new 49.60260 0.7087865 -3.61260045 0.02103158 ## 116 40.05 1 used 44.01777 0.5465195 -3.96776738 0.01250410 ## 117 45.00 1 new 49.60260 0.7087865 -4.60260045 0.02103158 ## 118 50.00 2 used 51.25061 0.8279109 -1.25060777 0.02869514 ## 119 49.75 0 new 42.36976 1.0651119 7.38023994 0.04749321 ## 120 47.00 1 used 44.01777 0.5465195 2.98223262 0.01250410 ## 121 56.00 2 new 56.83544 0.6764502 -0.83544085 0.01915635 ## 122 41.00 1 used 44.01777 0.5465195 -3.01776738 0.01250410 ## 123 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 124 34.99 1 used 44.01777 0.5465195 -9.02776738 0.01250410 ## 125 49.00 2 used 51.25061 0.8279109 -2.25060777 0.02869514 ## 126 61.00 2 new 56.83544 0.6764502 4.16455915 0.01915635 ## 127 62.89 2 new 56.83544 0.6764502 6.05455915 0.01915635 ## 128 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 129 64.95 2 new 56.83544 0.6764502 8.11455915 0.01915635 ## 130 36.99 0 used 36.78493 0.7065565 0.20507301 0.02089945 ## 131 44.00 1 used 44.01777 0.5465195 -0.01776738 0.01250410 ## 132 41.35 1 used 44.01777 0.5465195 -2.66776738 0.01250410 ## 133 37.00 0 used 36.78493 0.7065565 0.21507301 0.02089945 ## 134 58.98 2 new 56.83544 0.6764502 2.14455915 0.01915635 ## 135 39.00 0 used 36.78493 0.7065565 2.21507301 0.02089945 ## 136 40.70 1 used 44.01777 0.5465195 -3.31776738 0.01250410 ## 137 39.51 0 used 36.78493 0.7065565 2.72507301 0.02089945 ## 138 52.00 2 used 51.25061 0.8279109 0.74939223 0.02869514 ## 139 47.70 1 new 49.60260 0.7087865 -1.90260045 0.02103158 ## 140 38.76 0 used 36.78493 0.7065565 1.97507301 0.02089945 ## 141 54.51 2 new 56.83544 0.6764502 -2.32544085 0.01915635 ## .sigma .cooksd .std.resid ## 1 4.902339 1.161354e-03 0.402708933 ## 2 4.868399 8.712334e-03 -1.436710863 ## 3 4.892414 5.154337e-03 -0.848389768 ## 4 4.881308 9.612441e-03 -1.158579529 ## 5 4.750591 5.574926e-02 2.926332759 ## 6 4.899816 5.053659e-03 0.551419180 ## 7 4.905181 1.681147e-05 0.048608215 ## 8 4.899077 2.249739e-03 -0.587854989 ## 9 4.898517 1.591419e-03 0.614036807 ## 10 4.878184 6.403658e-03 1.231731888 ## 11 4.902639 9.463096e-04 -0.381259589 ## 12 4.904706 1.893237e-04 -0.170532281 ## 13 4.845644 2.169149e-02 -1.825361432 ## 14 4.904693 1.939387e-04 -0.172598235 ## 15 4.904866 8.464177e-05 -0.141610176 ## 16 4.840263 2.583436e-02 1.905485609 ## 17 4.898859 2.562308e-03 -0.598170028 ## 18 4.895349 3.974537e-03 -0.744993181 ## 19 4.904680 1.986092e-04 -0.174664189 ## 20 4.879726 1.018113e-02 -1.196202689 ## 21 4.899034 2.265580e-03 -0.589920943 ## 22 4.855017 1.829628e-02 1.676430592 ## 23 4.874681 1.115287e-02 -1.308872933 ## 24 4.900578 1.102520e-03 0.511087627 ## 25 4.902666 9.360817e-04 -0.379193635 ## 26 4.901254 1.588345e-03 -0.472475419 ## 27 4.904754 1.874384e-04 -0.162306590 ## 28 4.897361 3.144719e-03 0.664810290 ## 29 4.900072 2.074290e-03 -0.538200007 ## 30 4.875781 1.175148e-02 1.285147949 ## 31 4.879726 1.018113e-02 -1.196202689 ## 32 4.902124 7.357628e-04 -0.417513979 ## 33 4.902848 1.315656e-03 -0.365515142 ## 34 4.898356 1.629570e-03 -0.621353356 ## 35 4.904785 1.039625e-04 0.156942447 ## 36 4.825276 3.174557e-02 2.112264829 ## 37 4.905222 5.648698e-08 -0.003658274 ## 38 4.866239 1.422325e-02 1.478099008 ## 39 4.898043 2.628135e-03 -0.635371930 ## 40 4.895513 3.908619e-03 -0.738789386 ## 41 4.902867 5.591802e-04 -0.363980405 ## 42 4.884059 1.170136e-02 -1.090064849 ## 43 4.887380 6.523784e-03 -1.001045788 ## 44 4.904680 1.986092e-04 -0.174664189 ## 45 4.902124 7.357628e-04 -0.417513979 ## 46 4.899077 2.249739e-03 -0.587854989 ## 47 4.901493 1.492713e-03 0.458031070 ## 48 4.903987 4.946184e-04 0.263658603 ## 49 4.902261 7.030898e-04 0.408138446 ## 50 4.898186 2.575620e-03 0.628991915 ## 51 4.858711 1.853266e-02 -1.613896713 ## 52 4.856546 1.150293e-02 -1.650845158 ## 53 4.832389 6.760627e-02 2.016844445 ## 54 4.866054 1.561857e-02 1.481588208 ## 55 4.887793 6.965475e-03 -0.989423469 ## 56 4.893406 2.802864e-03 -0.814897814 ## 57 4.893260 2.837624e-03 0.819935167 ## 58 4.904754 1.874384e-04 -0.162306590 ## 59 4.879726 1.018113e-02 -1.196202689 ## 60 4.899077 2.249739e-03 -0.587854989 ## 61 4.870114 1.400524e-02 -1.402981909 ## 62 4.904194 3.768392e-04 0.240592564 ## 63 4.903908 5.261382e-04 0.271929772 ## 64 4.819876 4.687834e-02 2.181827236 ## 65 4.899077 2.249739e-03 -0.587854989 ## 66 4.893045 2.888492e-03 -0.827251716 ## 67 4.876193 4.052940e-02 1.276155547 ## 68 4.901375 2.130829e-03 -0.465166678 ## 69 4.900072 2.074290e-03 -0.538200007 ## 70 4.893644 2.746495e-03 -0.806661880 ## 71 4.855070 2.010496e-02 -1.675562463 ## 72 4.904693 1.939387e-04 -0.172598235 ## 73 4.855017 1.829628e-02 1.676430592 ## 74 4.901341 2.149892e-03 -0.467242751 ## 75 4.893260 2.837624e-03 0.819935167 ## 76 4.904101 4.491639e-04 0.251251850 ## 77 4.853667 2.053161e-02 1.698706389 ## 78 4.884908 8.116206e-03 1.068029769 ## 79 4.904897 7.741876e-05 -0.135433225 ## 80 4.902639 9.463096e-04 -0.381259589 ## 81 4.904152 2.543452e-04 0.245478742 ## 82 4.904806 1.218734e-03 -0.153165404 ## 83 4.902961 5.369254e-04 0.356663856 ## 84 4.866129 1.426303e-02 1.480164962 ## 85 4.904754 1.874384e-04 -0.162306590 ## 86 4.885435 7.957044e-03 1.054107431 ## 87 4.905151 2.893749e-05 0.063568128 ## 88 4.825276 3.174557e-02 2.112264829 ## 89 4.904442 1.853526e-04 -0.209556635 ## 90 4.904742 1.922448e-04 -0.164374382 ## 91 4.897178 4.453937e-03 0.672521687 ## 92 4.902261 7.030898e-04 0.408138446 ## 93 4.880253 9.971030e-03 -1.183795936 ## 94 4.890710 8.029235e-03 0.902965863 ## 95 4.901197 3.763354e-03 -0.475846028 ## 96 4.891531 5.008164e-03 -0.877088548 ## 97 4.864786 2.231340e-02 -1.505279580 ## 98 4.880180 5.932118e-03 -1.185514863 ## 99 4.905174 2.665668e-05 -0.052028020 ## 100 4.858308 1.710282e-02 -1.620831987 ## 101 4.834741 2.563232e-02 1.984257737 ## 102 4.871414 1.234172e-02 1.376867262 ## 103 4.884512 8.274103e-03 1.078368730 ## 104 4.900072 2.074290e-03 -0.538200007 ## 105 4.894442 5.966763e-03 0.778401443 ## 106 4.813060 2.168204e-02 -2.266481255 ## 107 4.898043 2.628135e-03 -0.635371930 ## 108 4.895349 3.974537e-03 -0.744993181 ## 109 4.875781 1.175148e-02 1.285147949 ## 110 4.903599 3.854926e-04 -0.302210897 ## 111 4.904027 6.625438e-04 0.259383029 ## 112 4.904888 1.227157e-04 0.137294864 ## 113 4.811529 7.605411e-02 2.285052621 ## 114 4.887082 4.300207e-03 1.009361659 ## 115 4.895294 3.996633e-03 -0.747061113 ## 116 4.893346 2.817046e-03 -0.816956798 ## 117 4.889096 6.487255e-03 -0.951786355 ## 118 4.904024 6.638336e-04 -0.259635386 ## 119 4.862490 3.978837e-02 1.547237493 ## 120 4.898517 1.591419e-03 0.614036807 ## 121 4.904693 1.939387e-04 -0.172598235 ## 122 4.898356 1.629570e-03 -0.621353356 ## 123 4.895349 3.974537e-03 -0.744993181 ## 124 4.843427 1.458352e-02 -1.858802502 ## 125 4.901341 2.149892e-03 -0.467242751 ## 126 4.892049 4.819157e-03 0.860378763 ## 127 4.877336 1.018587e-02 1.250844068 ## 128 4.895349 3.974537e-03 -0.744993181 ## 129 4.855017 1.829628e-02 1.676430592 ## 130 4.905191 1.279432e-05 0.042404838 ## 131 4.905222 5.648698e-08 -0.003658274 ## 132 4.899857 1.273496e-03 -0.549288929 ## 133 4.905187 1.407252e-05 0.044472630 ## 134 4.901733 1.277936e-03 0.443056056 ## 135 4.901493 1.492713e-03 0.458031070 ## 136 4.896922 1.969670e-03 -0.683122864 ## 137 4.899576 2.259209e-03 0.563488472 ## 138 4.904792 2.383611e-04 0.155579346 ## 139 4.902471 1.108535e-03 -0.393444786 ## 140 4.902257 1.186770e-03 0.408404057 ## 141 4.901119 1.502601e-03 -0.480425381# include interaction lm(totalPr ~ cond + duration + cond:duration, data=mario_kart)## ## Call: ## lm(formula = totalPr ~ cond + duration + cond:duration, data = mario_kart) ## ## Coefficients: ## (Intercept) condused duration ## 58.268 -17.122 -1.966 ## condused:duration ## 2.325# interaction plot ggplot(mario_kart, aes(x=duration, y=totalPr, color=cond)) + geom_point() + geom_smooth(method="lm", se=FALSE)
slr <- ggplot(mario_kart, aes(y = totalPr, x = duration)) + geom_point() + geom_smooth(method = "lm", se = 0) # model with one slope lm(totalPr ~ duration, data=mario_kart)## ## Call: ## lm(formula = totalPr ~ duration, data = mario_kart) ## ## Coefficients: ## (Intercept) duration ## 52.374 -1.317# plot with two slopes slr + aes(color=cond)
Chapter 3 - Multiple Regression
Adding a numerical explanatory variable - regressions with 2+ numerical variables:
- The mathematical expression is very simple - just another variable and coefficient added to the model
- The syntax for fitting the model is very simple also - Y ~ X1 + X2 rather than Y ~ X1
- The geometric (visualization) expression is nowhere near as simple
- ggplot does not handle a z-variable at all!
- ggplot does not handle a z-variable at all!
- Can run an approach known as “tiling the plane”
- grid <- babies %>%
- data_grid( gestation = seq_range(gestation, by = 1), age = seq_range(age, by = 1) )
- mod <- lm(bwt ~ gestation + age, data = babies)
- bwt_hats <- augment(mod, newdata = grid)
- data_space +
- geom_tile(data = bwt_hats, aes(fill = .fitted, alpha = 0.5)) +
- scale_fill_continuous(“bwt”, limits = range(babies$bwt))
- grid <- babies %>%
- Can also run the visuals using the plot_ly() call
- plot_ly(data = babies, z = ~bwt, x = ~gestation, y = ~age, opacity = 0.6) %>% # somewhat similar syntax to ggplot2
- add_markers(text = ~case, marker = list(size = 2)) %>% # draw points
- add_surface(x = ~x, y = ~y, z = ~plane, showscale = FALSE, cmax = 1, surfacecolor = color1, colorscale = col1) # draw plane
- plot_ly(data = babies, z = ~bwt, x = ~gestation, y = ~age, opacity = 0.6) %>% # somewhat similar syntax to ggplot2
Conditional interpretation of coefficients:
- Since a line cannot have two slopes, the coefficients from the multiple regression are defining a plane (two slopes) rather than a line
- The coefficients can instead be interpreted as meaning “given constant values for all the other predictor variables”
- Often referred to as “impact on Y of X1 while holding
constant”
- Often referred to as “impact on Y of X1 while holding
- Recall that each coefficient is in the same unit as the base data in the model; a coefficient being of larger magnitude does not mean that the coefficient is of larger importance
Adding a third (categorical) variable:
- Can add a third categorical variable to an existing model with two numerical variables already included
- Recall the geometry of the various models
- 1 numeric + 1 categorical: parallel slopes
- 2 numeric: plane
- 2 numeric + 1 categorical: parallel planes
- 1 numeric + 1 categorical: parallel slopes
- Can use the plotly::plot_ly() to visually represent the parallel planes
Higher dimensions:
- There are no mathematical or syntactical barrier to adding as many explanatory variables as needed in R
- There is, however, a geometric (visualization) barrier - parallel hyper-planes are not easy to visualize or interpret
- Can map to colors and project in to lower-dimensional spaces, but it does not always enhance understanding
- There is, however, a geometric (visualization) barrier - parallel hyper-planes are not easy to visualize or interpret
- Can use the dot operator to mean “everything” and the minus operator to mean “except”
- lm(Y ~ . - A) means regress Y and against everything, excluding A
- The key is to interpret each coefficient as “holding all other variables equal”
- The numeric variable coefficients are interpreted as slopes, while the factor variable coefficients are interpreted as changes to the intercept
- lm(Y ~ . - A) means regress Y and against everything, excluding A
Example code includes:
# Fit the model using duration and startPr (mod <- lm(totalPr ~ duration + startPr, data=mario_kart))## ## Call: ## lm(formula = totalPr ~ duration + startPr, data = mario_kart) ## ## Coefficients: ## (Intercept) duration startPr ## 51.030 -1.508 0.233# One method for visualizing a multiple linear regression model is to create a heatmap of the fitted values in the plane defined by the two explanatory variables # This heatmap will illustrate how the model output changes over different combinations of the explanatory variables # This is a multistep process # First, create a grid of the possible pairs of values of the explanatory variables. The grid should be over the actual range of the data present in each variable. We've done this for you and stored the result as a data frame called grid # Use augment() with the newdata argument to find the y-hat corresponding to the values in grid # Add these to the data_space plot by using the fill aesthetic and geom_tile() # add predictions to grid grid <- expand.grid(duration=1:10, startPr=seq(0.01, 69.95, by=0.01)) price_hats <- broom::augment(mod, newdata=grid) # tile the plane data_space <- mario_kart %>% filter(totalPr <= 75) %>% ggplot(aes(x=duration, y=startPr)) + geom_point(aes(col=totalPr)) data_space + geom_tile(data = price_hats, aes(fill=.fitted), alpha=0.5)
# An alternative way to visualize a multiple regression model with two numeric explanatory variables is as a plane in three dimensions. This is possible in R using the plotly package # We have created three objects that you will need # x: a vector of unique values of duration # y: a vector of unique values of startPr # plane: a matrix of the fitted values across all combinations of x and y # draw the 3D scatterplot p <- plotly::plot_ly(data = mario_kart, z = ~totalPr, x = ~duration, y = ~startPr, opacity = 0.6) %>% plotly::add_markers() # draw the plane x <- c(1, 1.13, 1.261, 1.391, 1.522, 1.652, 1.783, 1.913, 2.043, 2.174, 2.304, 2.435, 2.565, 2.696, 2.826, 2.957, 3.087, 3.217, 3.348, 3.478, 3.609, 3.739, 3.87, 4, 4.13, 4.261, 4.391, 4.522, 4.652, 4.783, 4.913, 5.043, 5.174, 5.304, 5.435, 5.565, 5.696, 5.826, 5.957, 6.087, 6.217, 6.348, 6.478, 6.609, 6.739, 6.87, 7, 7.13, 7.261, 7.391, 7.522, 7.652, 7.783, 7.913, 8.043, 8.174, 8.304, 8.435, 8.565, 8.696, 8.826, 8.957, 9.087, 9.217, 9.348, 9.478, 9.609, 9.739, 9.87, 10) y <- c(0.01, 1.024, 2.037, 3.051, 4.064, 5.078, 6.092, 7.105, 8.119, 9.133, 10.146, 11.16, 12.173, 13.187, 14.201, 15.214, 16.228, 17.242, 18.255, 19.269, 20.282, 21.296, 22.31, 23.323, 24.337, 25.351, 26.364, 27.378, 28.391, 29.405, 30.419, 31.432, 32.446, 33.46, 34.473, 35.487, 36.5, 37.514, 38.528, 39.541, 40.555, 41.569, 42.582, 43.596, 44.609, 45.623, 46.637, 47.65, 48.664, 49.678, 50.691, 51.705, 52.718, 53.732, 54.746, 55.759, 56.773, 57.787, 58.8, 59.814, 60.827, 61.841, 62.855, 63.868, 64.882, 65.896, 66.909, 67.923, 68.936, 69.95) grid <- expand.grid(duration=x, startPr=y) predPr <- broom::augment(mod, newdata=grid) plane <- matrix(data=predPr$.fitted, nrow=70, ncol=70, byrow=FALSE) p <- p %>% plotly::add_surface(x = ~x, y = ~y, z = ~plane, showscale = FALSE) # Commented due to inability to use in html # p # draw the 3D scatterplot # p <- plotly::plot_ly(data = mario_kart, z = ~totalPr, x = ~duration, y = ~startPr, opacity = 0.6) %>% # plotly::add_markers(color = ~cond) # draw two planes # p %>% # add_surface(x = ~x, y = ~y, z = ~plane0, showscale = FALSE) %>% # add_surface(x = ~x, y = ~y, z = ~plane1, showscale = FALSE)
Chapter 4 - Logistic Regression
What is logistic regression?
- When graphing a categorical variable as the response, geom_jitter() can be much more helpful than geom_point()
- With a categorical variable as a response, it is important to convert the variable to a numeric
- A straight linear regression with a numeric predictor and a categorical response can make non-sensical predictions
- Might predict a -20% chance of death or a 120% chance of death even for reasonable values of the predictor variable, for example
- Might predict a -20% chance of death or a 120% chance of death even for reasonable values of the predictor variable, for example
- The GLM (Generalized Linear Model) formalizes a modeling technique for predicting categorical variables
- The logistic regression (logit) is frequently used to model a binary response
- The basic idea is that a link function is added - logit in the case of the logistic regression
- The logistic regression (logit) is frequently used to model a binary response
- Fitting a GLM in R is very similar to fitting an LM
- glm(myFactor ~ myNumeric, family=binomial) # will run the logistic regression
- glm(myFactor ~ myNumeric, family=binomial) # will run the logistic regression
- Note that the mathematical model is now:
- log(y / (1−y))=β0+β1⋅x+ϵ
- where ϵ is the error term
- log(y / (1−y))=β0+β1⋅x+ϵ
Visualizing logistic regression:
- Can use geom_smooth(method=“glm”, se=0, color=“red”, method.args=list(family=“binomial”)) to add a logit smooth to a plot
- The logistic regression can never precisely predict the categorical variable, since the data are all 0 or 1 while the logit is between 0 and 1
- Can bin the underlying data, look at aggregate probabilities (mean x, mean y) by bin, and compare with the logit outcomes
- Here we are plotting y as a function of x, where that function is
- y = exp(β0+β1⋅x) / (1+exp(β0+β1⋅x))
Three scales approach to visualization:
- Generating the predicted values comes from myGLM %>% augment(type.predict=“response”) %>% mutate(y_hat = .fitted)
- Can no longer interpret the coefficients as slopes
- odds = y-hat / (1 - y-hat) = exp(B0 + B1 * X1)
- log(odds) = log( (y-hat / (1 - y-hat)) ) = B0 + B1 * X1
- So, the coefficients do represent the slope of the log-odds function
- odds = y-hat / (1 - y-hat) = exp(B0 + B1 * X1)
- Comparison of the three potential scales for visualizing a logistic regression
- Probability - positives (intuitive and easy to interpret scale) and negatives (non-linear and challenging to interpret function)
- Odds - a bit of a middle ground between Probability and Log-Odds
- Log-Odds - positive (linear, easy-to-interpret function) and negative (very hard to interpret scale)
- Probability - positives (intuitive and easy to interpret scale) and negatives (non-linear and challenging to interpret function)
- Can also calculate the odds-ratio, as well as the change in odds-ratio with respect to a key variable
- OR = odds(y | x + 1) / odds(y | x) = exp(B1)
- If the exp(coef(myGLM)) has a coefficient greater than 1, then the odds increase with that variable; otherwise, they decrease
- OR = odds(y | x + 1) / odds(y | x) = exp(B1)
Using a logistical model - objective is to gain better understanding in to the underlying process:
- Can augment the GLM to add a variable for heart transplant - will be very meaningful!
- Note that the default for the augment() function is to give back .fitted that represents log-odds scale
- Can instead set the type.predict = “response” to get back the normal probability scale
- Can instead set the type.predict = “response” to get back the normal probability scale
- Additional goal is to make out-of-sample predictions based on the results of the model
- Since the study was performed in 1973, it is likely that the coefficients have changed
- Since the study was performed in 1973, it is likely that the coefficients have changed
- Can also convert the predictions by rounding the probabilities (less than 0.5 is death, greater than 0.5 is life) and then running the confusion matrix
- Can experiment with different rounding thresholds than 0.5, and see whether the confusion matrix is improved
Example code includes:
# To see this in action, we'll fit a linear regression model to data about 55 students who applied to medical school # We want to understand how their undergraduate GPAGPA relates to the probability they will be accepted by a particular school (Acceptance) # The medical school acceptance data is loaded in your workspace as MedGPA # scatterplot with jitter # tmpSAT <- readr::read_csv("./RInputFiles/SAT.csv") tmpAccept <- c(0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0) tmpGPA <- c(3.62, 3.84, 3.23, 3.69, 3.38, 3.72, 3.89, 3.34, 3.71, 3.89, 3.97, 3.49, 3.77, 3.61, 3.3, 3.54, 3.65, 3.54, 3.25, 3.89, 3.71, 3.77, 3.91, 3.88, 3.68, 3.56, 3.44, 3.58, 3.4, 3.82, 3.62, 3.09, 3.89, 3.7, 3.24, 3.86, 3.54, 3.4, 3.87, 3.14, 3.37, 3.38, 3.62, 3.94, 3.37, 3.36, 3.97, 3.04, 3.29, 3.67, 2.72, 3.56, 3.48, 2.8, 3.44) MedGPA <- data.frame(Acceptance=tmpAccept, GPA=tmpGPA) str(MedGPA)## 'data.frame': 55 obs. of 2 variables: ## $ Acceptance: num 0 1 1 1 1 1 1 0 1 1 ... ## $ GPA : num 3.62 3.84 3.23 3.69 3.38 3.72 3.89 3.34 3.71 3.89 ...data_space <- ggplot(MedGPA, aes(x=GPA, y=Acceptance)) + geom_jitter(width = 0, height = 0.05, alpha = 0.5) # linear regression line data_space + geom_smooth(method="lm", se=FALSE)
# filter MedGPA_middle <- MedGPA %>% filter(GPA >= 3.375, GPA <= 3.77) # scatterplot with jitter data_space <- ggplot(MedGPA_middle, aes(x=GPA, y=Acceptance)) + geom_jitter(width = 0, height = 0.05, alpha = 0.5) # linear regression line data_space + geom_smooth(method="lm", se=FALSE)
# fit model (mod <- glm(Acceptance ~ GPA, data = MedGPA, family = binomial))## ## Call: glm(formula = Acceptance ~ GPA, family = binomial, data = MedGPA) ## ## Coefficients: ## (Intercept) GPA ## -19.207 5.454 ## ## Degrees of Freedom: 54 Total (i.e. Null); 53 Residual ## Null Deviance: 75.79 ## Residual Deviance: 56.84 AIC: 60.84# scatterplot with jitter data_space <- ggplot(MedGPA, aes(x=GPA, y=Acceptance)) + geom_jitter(width=0, height=0.05, alpha = .5) # add logistic curve data_space + geom_smooth(method="glm", se=FALSE, method.args=structure(list(family = "binomial"), .Names = "family"))
# We have created a data.frame called MedGPA_binned that aggregates the original data into separate bins for each 0.25 of GPA. It also contains the fitted values from the logistic regression model MedGPA$bin <- round(MedGPA$GPA*4, 0) / 4 str(MedGPA)## 'data.frame': 55 obs. of 3 variables: ## $ Acceptance: num 0 1 1 1 1 1 1 0 1 1 ... ## $ GPA : num 3.62 3.84 3.23 3.69 3.38 3.72 3.89 3.34 3.71 3.89 ... ## $ bin : num 3.5 3.75 3.25 3.75 3.5 3.75 4 3.25 3.75 4 ...MedGPA_binned <- MedGPA %>% group_by(bin) %>% summarize(mean_GPA=mean(GPA), acceptance_rate=mean(Acceptance), ct=n()) MedGPA_binned## # A tibble: 6 x 4 ## bin mean_GPA acceptance_rate ct ## <dbl> <dbl> <dbl> <int> ## 1 2.75 2.76 0 2 ## 2 3.00 3.06 0 2 ## 3 3.25 3.29 0.200 10 ## 4 3.50 3.51 0.556 18 ## 5 3.75 3.75 0.643 14 ## 6 4.00 3.91 1.00 9# binned points and line data_space <- ggplot(MedGPA_binned, aes(x=mean_GPA, y=acceptance_rate)) + geom_point() + geom_line() # augmented model MedGPA_plus <- broom::augment(mod, type.predict="response") # logistic model on probability scale data_space + geom_line(data = MedGPA_plus, aes(x=GPA, y=.fitted), color = "red")
# The MedGPA_binned data frame contains the data for each GPA bin, while the MedGPA_plus data frame records the original observations after being augment()-ed by mod # compute odds for bins MedGPA_binned <- MedGPA_binned %>% mutate(odds = acceptance_rate / (1 - acceptance_rate)) # plot binned odds data_space <- ggplot(MedGPA_binned, aes(x=mean_GPA, y=odds)) + geom_point() + geom_line() # compute odds for observations MedGPA_plus <- MedGPA_plus %>% mutate(odds_hat = .fitted / (1 - .fitted)) # logistic model on odds scale data_space + geom_line(data=MedGPA_plus, aes(x=GPA, y=odds_hat), color = "red")
# compute log odds for bins MedGPA_binned <- MedGPA_binned %>% mutate(log_odds = log(acceptance_rate / (1 - acceptance_rate))) # plot binned log odds data_space <- ggplot(MedGPA_binned, aes(x=mean_GPA, y=log_odds)) + geom_point() + geom_line() # compute log odds for observations MedGPA_plus <- MedGPA_plus %>% mutate(log_odds_hat = log(.fitted / (1 - .fitted))) # logistic model on log odds scale data_space + geom_line(data=MedGPA_plus, aes(x=GPA, y=log_odds_hat), color = "red")
# create new data frame new_data <- data.frame(GPA = 3.51) # make predictions broom::augment(mod, newdata=new_data, type.predict="response")## GPA .fitted .se.fit ## 1 3.51 0.4844099 0.08343193# data frame with binary predictions tidy_mod <- broom::augment(mod, type.predict="response") %>% mutate(Acceptance_hat = round(.fitted)) # confusion matrix tidy_mod %>% select(Acceptance, Acceptance_hat) %>% table()## Acceptance_hat ## Acceptance 0 1 ## 0 16 9 ## 1 6 24
Chapter 5 - Case Study: Italian Restaurants in NYC
Italian restaurants in NYC - factors that contribute to the price of a meal:
- Will use the numeric data available from Zagat, using the scores that range from 0-30
- The dataset “nyc” has Price and several other (potential) explanatory variables
- Can run pairs(nyc) to see some basic relationships in the variables for this dataset
Incorporating another variable:
- Fifth Island divides Manhattan, and historically the East side has been the pricier side of the island
- Question is whether, all else equal, there is an East-side premium in the restaurant pricing
- Question is whether, all else equal, there is an East-side premium in the restaurant pricing
- Additional question is whether people are willing to pay more for better service, holding the quality of the food constant
Higher dimensions - adding the décor dimension to the existing Zagat analysis:
- Objective is to predict Price, with explanatory variables including Food, Service, Décor, and East (categorical)
- Can introduce collinearity problems since many of the variables are potentially correlated with each other
- Can also have collinearity across multiple variable, such as Y ~ A + B + C when A = B + C
- The main problem with collinearity is that it makes the coefficients extremely unreliable (non-robust)
- Note that multi-collinearity does NOT hurt the explanatory power of the model - R-squared is still OK
- Can also have collinearity across multiple variable, such as Y ~ A + B + C when A = B + C
Wrap up:
- Mathematical, Geometric, Syntactical modeling
- Focus was on descriptive statistics rather than inferential statistics
- Prior to inferential statistics, only conclusions and descriptions about the sample can be made
Example code includes:
nyc <- readr::read_csv("./RInputFiles/nyc.csv")## Parsed with column specification: ## cols( ## Case = col_integer(), ## Restaurant = col_character(), ## Price = col_integer(), ## Food = col_integer(), ## Decor = col_integer(), ## Service = col_integer(), ## East = col_integer() ## )str(nyc, give.attr=FALSE)## Classes 'tbl_df', 'tbl' and 'data.frame': 168 obs. of 7 variables: ## $ Case : int 1 2 3 4 5 6 7 8 9 10 ... ## $ Restaurant: chr "Daniella Ristorante" "Tello's Ristorante" "Biricchino" "Bottino" ... ## $ Price : int 43 32 34 41 54 52 34 34 39 44 ... ## $ Food : int 22 20 21 20 24 22 22 20 22 21 ... ## $ Decor : int 18 19 13 20 19 22 16 18 19 17 ... ## $ Service : int 20 19 18 17 21 21 21 21 22 19 ... ## $ East : int 0 0 0 0 0 0 0 1 1 1 ...# Price by Food plot ggplot(nyc, aes(x=Food, y=Price)) + geom_point()
# Price by Food model lm(Price ~ Food, data=nyc)## ## Call: ## lm(formula = Price ~ Food, data = nyc) ## ## Coefficients: ## (Intercept) Food ## -17.832 2.939# fit model lm(Price ~ Food + Service, data=nyc)## ## Call: ## lm(formula = Price ~ Food + Service, data = nyc) ## ## Coefficients: ## (Intercept) Food Service ## -21.159 1.495 1.704# draw 3D scatterplot # p <- plot_ly(data = nyc, z = ~Price, x = ~Food, y = ~Service, opacity = 0.6) %>% # add_markers() # draw a plane # p %>% # add_surface(x = ~x, y = ~y, z = ~plane, showscale = FALSE) # Price by Food and Service and East lm(Price ~ Food + Service + East, data=nyc)## ## Call: ## lm(formula = Price ~ Food + Service + East, data = nyc) ## ## Coefficients: ## (Intercept) Food Service East ## -20.8155 1.4863 1.6647 0.9649# draw 3D scatterplot # p <- plot_ly(data = nyc, z = ~Price, x = ~Food, y = ~Service, opacity = 0.6) %>% # add_markers(color = ~factor(East)) # draw two planes # p %>% # add_surface(x = ~x, y = ~y, z = ~plane0, showscale = FALSE) %>% # add_surface(x = ~x, y = ~y, z = ~plane1, showscale = FALSE)
Forecasting Using R
Chapter 1 - Exploring and Visualizing Time Series in R
Introduction and overview:
- Course contents include
- Visualizing time series
- Simple benchmarks for forecasting
- Exponential smoothing and ARIMA
- Advanced forecasting methods
- Measuring forecast accuracy
- Choosing the best methods
- Visualizing time series
- Hyndman book “Forecasting Principles and Practice” is freely available online
- Data in this book will be “regularly spaced” - yearly, hourly, etc.
- Forecasting is the process of projecting the time-series data forward, and including prediction intervals
- Trend, seasonality, other features
- Trend, seasonality, other features
- A time series can be thought of as a vector or matrix of numbers along with some information about what times those numbers were recorded
- This information is stored in a ts object in R (in most exercises, you will use time series that are part of existing packages)
- If you want to work with your own data, you need to know how to create a ts object in R
- This information is stored in a ts object in R (in most exercises, you will use time series that are part of existing packages)
- ts(data, start, frequency, …)
- frequency is set to 4 if the data are quarterly
- start is set to the form c(year, period) to indicate the time of the first observation
- Here, January corresponds with period 1; likewise, a start date in April would refer to 2, July to 3, and October to 4. Thus, period corresponds to the quarter of the year
- frequency is set to 4 if the data are quarterly
- You can use the autoplot() function to produce a time plot of the data with or without facets, or panels that display different subsets of data:
- autoplot(usnim_2002, facets = FALSE)
- autoplot(usnim_2002, facets = FALSE)
- To find the number of observations per unit time, use frequency()
- frequency(usnim_2002)
- frequency(usnim_2002)
- Along with time plots, there are other useful ways of plotting data to emphasize seasonal patterns and show changes in these patterns over time
- A seasonal plot is similar to a time plot except that the data are plotted against the individual “seasons” in which the data were observed
- You can create one using the ggseasonplot() function the same way you do with autoplot()
- An interesting variant of a season plot uses polar coordinates, where the time axis is circular rather than horizontal; to make one, simply add a polar argument and set it to TRUE
- A subseries plot comprises mini time plots for each season. Here, the mean for each season is shown as a blue horizontal line
- A seasonal plot is similar to a time plot except that the data are plotted against the individual “seasons” in which the data were observed
- One way of splitting a time series is by using the window() function, which extracts a subset from the object x observed between the times start and end
- window(x, start = NULL, end = NULL)
Trends, seasonality, and cyclicity:
- Several types of patterns in time series are so common that they get named
- Trends are long-term patterns of increase or decrease in the data
- Seasonality is a periodic pattern that recurs in the data
- Cyclic is a pattern of the data rising or falling (typically over 2+ years) that are not of a fixed period - example being a business cycle
- Trends are long-term patterns of increase or decrease in the data
- Be cautious about cycles vs. trends, and be sure that the window is long enough to make the distinction
- The lynx population in Canada is an example of a very dramatic cyclic pattern (too many lynx -> not enough food per lynx > most lynx starve to death -> plenty of food per survivor -> breeding -> too many lynx)
- Another way to look at time series data is to plot each observation against another observation that occurred some time previously by using gglagplot()
- For example, you could plot yt against yt-1. This is called a lag plot because you are plotting the time series against lags of itself
- The correlations associated with the lag plots form what is called the autocorrelation function (ACF). The ggAcf() function produces ACF plots.
- For example, you could plot yt against yt-1. This is called a lag plot because you are plotting the time series against lags of itself
- When data are either seasonal or cyclic, the ACF will peak around the seasonal lags or at the average cycle length
White noise - simply a time series of independently and identically distributed (iid) data:
- White noise serves as the basis for many types of time series modeling
- The samping distributions for white noise ACF are well known, and the dashed lines on ACF plots are frequently the 95% interval for these
- ACF that are outside the dashed lines are frequently suggestive that there is information beyond white-noise
- The samping distributions for white noise ACF are well known, and the dashed lines on ACF plots are frequently the 95% interval for these
- The Ljung-Box test considers the first h autocorrelation values together
- A significant p-value indicates that these data are likely NOT white noise
- Appears to be a modified implementation of chi-squared
- Box.test(pigs, lag = 24, fitdf = 0, type = “Ljung”)
- A significant p-value indicates that these data are likely NOT white noise
- There is a well-known result in economics called the “Efficient Market Hypothesis” that states that asset prices reflect all available information
- A consequence of this is that the daily changes in stock prices should behave like white noise (ignoring dividends, interest rates and transaction costs)
- The consequence for forecasters is that the best forecast of the future price is the current price
- A consequence of this is that the daily changes in stock prices should behave like white noise (ignoring dividends, interest rates and transaction costs)
Example code includes:
library(forecast) # Read the data from Excel into R mydata <- readxl::read_excel("./RInputFiles/exercise1.xlsx") # Look at the first few lines of mydata head(mydata)## # A tibble: 6 x 4 ## X__1 Sales AdBudget GDP ## <chr> <dbl> <dbl> <dbl> ## 1 Mar-81 1020 659 252 ## 2 Jun-81 889 589 291 ## 3 Sep-81 795 512 291 ## 4 Dec-81 1004 614 292 ## 5 Mar-82 1058 647 279 ## 6 Jun-82 944 602 254# Create a ts object called myts myts <- ts(mydata[, -1], start = c(1981, 1), frequency = 4) # Plot the data with facetting autoplot(myts, facets = TRUE)
# Plot the data without facetting autoplot(myts, facets = FALSE)
# Plot the three series data(gold, package="forecast") data(woolyrnq, package="forecast") data(gas, package="forecast") str(gold)## Time-Series [1:1108] from 1 to 1108: 306 300 303 297 304 ...str(woolyrnq)## Time-Series [1:119] from 1965 to 1994: 6172 6709 6633 6660 6786 ...str(gas)## Time-Series [1:476] from 1956 to 1996: 1709 1646 1794 1878 2173 ...autoplot(gold)
autoplot(woolyrnq)
autoplot(gas)
# Find the outlier in the gold series goldoutlier <- which.max(gold) # Look at the seasonal frequencies of the three series frequency(gold)## [1] 1frequency(woolyrnq)## [1] 4frequency(gas)## [1] 12# In this exercise, you will load the fpp2 package and use two of its datasets # a10 contains monthly sales volumes for anti-diabetic drugs in Australia. In the plots, can you see which month has the highest sales volume each year? What is unusual about the results in March and April 2008? # ausbeer which contains quarterly beer production for Australia. What is happening to the beer production in Quarter 4? # Load the fpp2 package # library(fpp2) data(a10, package="fpp2") data(ausbeer, package="fpp2") str(a10)## Time-Series [1:204] from 1992 to 2008: 3.53 3.18 3.25 3.61 3.57 ...str(ausbeer)## Time-Series [1:218] from 1956 to 2010: 284 213 227 308 262 228 236 320 272 233 ...# Create plots of the a10 data autoplot(a10)
forecast::ggseasonplot(a10)
# Produce a polar coordinate season plot for the a10 data forecast::ggseasonplot(a10, polar = TRUE)
# Restrict the ausbeer data to start in 1992 beer <- window(ausbeer, start=1992) # Make plots of the beer data autoplot(beer)
forecast::ggsubseriesplot(beer)
# In this exercise, you will work with the pre-loaded oil data (available in the package fpp2), which contains the annual oil production in Saudi Arabia from 1965-2013 (measured in millions of tons). # Create an autoplot of the oil data data(oil, package="fpp2") str(oil)## Time-Series [1:49] from 1965 to 2013: 111 131 141 154 163 ...autoplot(oil)
# Create a lag plot of the oil data forecast::gglagplot(oil)
# Create an ACF plot of the oil data library(forecast) ggAcf(oil)
# You will investigate this phenomenon by plotting the annual sunspot series (which follows the solar cycle of approximately 10-11 years) in sunspot.year # and the daily traffic to the Hyndsight blog (which follows a 7-day weekly pattern) in hyndsight. Both objects have been loaded into your workspace. # Plot the annual sunspot numbers data(sunspot.year) str(sunspot.year)## Time-Series [1:289] from 1700 to 1988: 5 11 16 23 36 58 29 20 10 8 ...autoplot(sunspot.year)
ggAcf(sunspot.year)
# Plot the traffic on the Hyndsight blog data(hyndsight, package="fpp2") str(hyndsight)## Time-Series [1:365] from 1.43 to 53.4: 1157 1118 1310 874 890 1437 1263 1187 1506 1448 ...autoplot(hyndsight)
ggAcf(hyndsight)
# You can test this hypothesis by looking at the goog series, which contains the closing stock price for Google over 1000 trading days ending on February 13, 2017. This data has been loaded into your workspace. # Plot the original series data(goog, package="fpp2") str(goog)## Time-Series [1:1000] from 1 to 1000: 393 393 397 398 400 ...autoplot(goog)
# Plot the differenced series autoplot(diff(goog))
# ACF of the differenced series ggAcf(diff(goog))
# Ljung-Box test of the differenced series Box.test(diff(goog), lag = 10, type = "Ljung")## ## Box-Ljung test ## ## data: diff(goog) ## X-squared = 13.123, df = 10, p-value = 0.2169
Chapter 2 - Benchmark methods and forecast accuracy
Forecasts and potential futures:
- Can build up a distribution of potential futures based on multiple simulations (NOT forecasts - each is a simulated future)
- The point forecast (“forecast”) is the average of these multiple simulations
- The shaded regions are typically the 80% / 95% CI intervals around the point forecasts
- The point forecast (“forecast”) is the average of these multiple simulations
- The very simplest forecasting method is to use the most recent observation; this is called a naive forecast and can be implemented in a namesake function
- This is the best that can be done for many time series including most stock price data
- Even if it is not a good forecasting method, it provides a useful benchmark for other forecasting methods
- This is the best that can be done for many time series including most stock price data
- This is implemented in the snaive() function, meaning, seasonal naive
- For both forecasting methods, you can set the second argument h, which specifies the number of values you want to forecast
- naive(y, h = 10)
- snaive(y, h = 2 * frequency(x))
- For both forecasting methods, you can set the second argument h, which specifies the number of values you want to forecast
Fitted values and residuals:
- Frequent validation method is to forecast using historical data ON TO data that you already know
- Allows for both fitted values and residuals to be calculated - residuals should look like white noise in a good forecast
- Sometimes, there is some “cheating”, since the observed data point was also used in building the forecast model
- Allows for both fitted values and residuals to be calculated - residuals should look like white noise in a good forecast
- Four basic assumptions that are made about the residuals
- Residuals should be uncorrelated (essential)
- Residuals should have zero mean (essential - otherwise the model is biased)
- Residuals have constant variance (convenirnce)
- Residuals are normally distributed (convenience)
- Residuals should be uncorrelated (essential)
- Basically, the request is for the residuals to be Gaussian white noise, which can be checked using checkresiduals(myFCModel)
- Checking residuals is an essential step prior to moving forward with the forecast
Training and test sets help to validate the forecasting methodology:
- Forecasts can be built using the training set, and then applied to the test set for validation
- Checking the forecasts on the test set helps solve the issues of over-fitting
- Forecast “errors” are the differences between the forecast values and the observed values in the test set
- In contrast, residuals are the differences in fitted and actual on the training set
- Further, residuals are based only on the one-step forecast (???)
- In contrast, residuals are the differences in fitted and actual on the training set
- There are many metrics used for assessing the quality of the forecast
- Mean Absolute Error (MAE)
- Mean Squared Error (MSE)
- Mean Absolute Percentage Error (MAPE)
- Mean Absolute Scaled Error (MASE) = MAE / Q where Q is a scaling constant
- Mean Absolute Error (MAE)
- The accuracy() function calculates all of the key measures above - training set based on residuals, and test set based on errors
- Useful for comparing multiple forecast models on the same data
- Useful for comparing multiple forecast models on the same data
- One function that can be used to create training and test sets is subset.ts(), which returns a subset of a time series where the optional start and end arguments are specified using index values.
To subset observations from 101 to 500
train <- subset.ts(x, start = 101, end = 500, …)
To subset the first 500 observations
train <- subset.ts(x, end = 500, …)
- The function meanf(), which gives forecasts equal to the mean of all observations
Time series cross-validation attempts to solve some of the problems related to test-train:
- A general process is to make each test set a single point following a training set of variable lengths
- Analogous to cross-validation in non-time-series problems
- Analogous to cross-validation in non-time-series problems
- The tsCV() function does a lot of the work behind the scenes
- myE <- tsCV(myTS, forecastfunction = naïve, h=1)
- mean(myE ** 2, na.rm=TRUE)
- myE <- tsCV(myTS, forecastfunction = naïve, h=1)
- Generally, the MSE will increase as the window (h) increases
Example code includes:
# Use naive() to forecast the goog series fcgoog <- naive(goog, h=20) # Plot and summarize the forecasts autoplot(fcgoog)
summary(fcgoog)## ## Forecast method: Naive method ## ## Model Information: ## Call: naive(y = goog, h = 20) ## ## Residual sd: 8.7285 ## ## Error measures: ## ME RMSE MAE MPE MAPE MASE ## Training set 0.4212612 8.734286 5.829407 0.06253998 0.9741428 1 ## ACF1 ## Training set 0.03871446 ## ## Forecasts: ## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 ## 1001 813.67 802.4765 824.8634 796.5511 830.7889 ## 1002 813.67 797.8401 829.4999 789.4602 837.8797 ## 1003 813.67 794.2824 833.0576 784.0192 843.3208 ## 1004 813.67 791.2831 836.0569 779.4322 847.9078 ## 1005 813.67 788.6407 838.6993 775.3910 851.9490 ## 1006 813.67 786.2518 841.0882 771.7374 855.6025 ## 1007 813.67 784.0549 843.2850 768.3777 858.9623 ## 1008 813.67 782.0102 845.3298 765.2505 862.0895 ## 1009 813.67 780.0897 847.2503 762.3133 865.0266 ## 1010 813.67 778.2732 849.0667 759.5353 867.8047 ## 1011 813.67 776.5456 850.7944 756.8931 870.4469 ## 1012 813.67 774.8948 852.4452 754.3684 872.9715 ## 1013 813.67 773.3115 854.0285 751.9470 875.3930 ## 1014 813.67 771.7880 855.5520 749.6170 877.7230 ## 1015 813.67 770.3180 857.0220 747.3688 879.9711 ## 1016 813.67 768.8962 858.4437 745.1944 882.1455 ## 1017 813.67 767.5183 859.8217 743.0870 884.2530 ## 1018 813.67 766.1802 861.1597 741.0407 886.2993 ## 1019 813.67 764.8789 862.4610 739.0505 888.2895 ## 1020 813.67 763.6114 863.7286 737.1120 890.2280# Use snaive() to forecast the ausbeer series fcbeer <- snaive(ausbeer, h=16) # Plot and summarize the forecasts autoplot(fcbeer)
summary(fcbeer)## ## Forecast method: Seasonal naive method ## ## Model Information: ## Call: snaive(y = ausbeer, h = 16) ## ## Residual sd: 19.1207 ## ## Error measures: ## ME RMSE MAE MPE MAPE MASE ACF1 ## Training set 3.098131 19.32591 15.50935 0.838741 3.69567 1 0.01093868 ## ## Forecasts: ## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 ## 2010 Q3 419 394.2329 443.7671 381.1219 456.8781 ## 2010 Q4 488 463.2329 512.7671 450.1219 525.8781 ## 2011 Q1 414 389.2329 438.7671 376.1219 451.8781 ## 2011 Q2 374 349.2329 398.7671 336.1219 411.8781 ## 2011 Q3 419 383.9740 454.0260 365.4323 472.5677 ## 2011 Q4 488 452.9740 523.0260 434.4323 541.5677 ## 2012 Q1 414 378.9740 449.0260 360.4323 467.5677 ## 2012 Q2 374 338.9740 409.0260 320.4323 427.5677 ## 2012 Q3 419 376.1020 461.8980 353.3932 484.6068 ## 2012 Q4 488 445.1020 530.8980 422.3932 553.6068 ## 2013 Q1 414 371.1020 456.8980 348.3932 479.6068 ## 2013 Q2 374 331.1020 416.8980 308.3932 439.6068 ## 2013 Q3 419 369.4657 468.5343 343.2438 494.7562 ## 2013 Q4 488 438.4657 537.5343 412.2438 563.7562 ## 2014 Q1 414 364.4657 463.5343 338.2438 489.7562 ## 2014 Q2 374 324.4657 423.5343 298.2438 449.7562# Check the residuals from the naive forecasts applied to the goog series goog %>% naive() %>% checkresiduals()
## ## Ljung-Box test ## ## data: Residuals from Naive method ## Q* = 13.123, df = 10, p-value = 0.2169 ## ## Model df: 0. Total lags used: 10# Do they look like white noise (TRUE or FALSE) googwn <- TRUE # Check the residuals from the seasonal naive forecasts applied to the ausbeer series ausbeer %>% snaive() %>% checkresiduals()
## ## Ljung-Box test ## ## data: Residuals from Seasonal naive method ## Q* = 60.535, df = 8, p-value = 3.661e-10 ## ## Model df: 0. Total lags used: 8# Do they look like white noise (TRUE or FALSE) beerwn <- FALSE # The pre-loaded time series gold comprises daily gold prices for 1108 days. Here, you'll use the first 1000 days as a training set, and compute forecasts for the remaining 108 days # Create the training data as train train <- subset(gold, end = 1000) # Compute naive forecasts and save to naive_fc naive_fc <- naive(train, h = 108) # Compute mean forecasts and save to mean_fc mean_fc <- meanf(train, h = 108) # Use accuracy() to compute RMSE statistics accuracy(naive_fc, gold)## ME RMSE MAE MPE MAPE MASE ## Training set 0.09161392 6.33977 3.158386 0.01662141 0.794523 1.000000 ## Test set -6.53834951 15.84236 13.638350 -1.74622688 3.428789 4.318139 ## ACF1 Theil's U ## Training set -0.3098928 NA ## Test set 0.9793153 5.335899accuracy(mean_fc, gold)## ME RMSE MAE MPE MAPE MASE ## Training set -4.239671e-15 59.17809 53.63397 -2.390227 14.230224 16.981449 ## Test set 1.319363e+01 19.55255 15.66875 3.138577 3.783133 4.960998 ## ACF1 Theil's U ## Training set 0.9907254 NA ## Test set 0.9793153 6.123788# Assign one of the two forecasts as bestforecasts # bestforecasts <- naive_fc # Here, you will use the Melbourne quarterly visitor numbers (vn[, "Melbourne"]) to create three different training sets, omitting the last 1, 2 and 3 years, respectively # Inspect the pre-loaded vn data in your console before beginning the exercise # This will help you determine the correct value to use for the keyword h (which specifies the number of values you want to forecast) in your forecasting methods melData <- c(4.865, 4.113, 4.422, 5.171, 5.55, 4.009, 3.986, 3.839, 5.8, 4.229, 4.157, 4.627, 5.691, 4.601, 4.742, 5.733, 5.397, 3.884, 4.996, 5.304, 5.222, 4.765, 4.146, 4.717, 4.88, 4.868, 4.182, 4.214, 5.438, 3.87, 4.394, 4.404, 5.716, 5.291, 4.19, 4.712, 4.709, 4.489, 4.698, 5.193, 5.216, 4.215, 5.042, 5.089, 4.688, 4.393, 4.626, 4.88, 4.844, 4.437, 4.833, 4.622, 5.164, 4.504, 4.976, 4.508, 4.759, 4.835, 5.009, 5.693, 5.224, 4.82, 4.688, 4.918, 5.936, 5.44, 5.134, 5.993, 6.654, 5.342, 5.471, 5.812) sydData <- c(7.319, 6.13, 6.284, 6.384, 6.602, 5.674, 5.715, 6.564, 6.602, 5.398, 7.172, 8.474, 7.012, 6.388, 6.073, 6.196, 5.633, 5.779, 5.869, 6.002, 6.202, 5.321, 5.161, 5.737, 6.168, 5.709, 5.057, 5.362, 5.902, 4.496, 5.093, 5.253, 6.832, 5.67, 5.008, 5.773, 6.529, 4.911, 4.784, 5.844, 6.252, 5.034, 5.263, 4.714, 5.362, 4.769, 4.125, 5.263, 6, 4.283, 5.256, 5.357, 6.194, 5.102, 5.596, 5.066, 6.684, 4.697, 5.366, 5.075, 5.499, 4.867, 5.71, 6.198, 6.416, 5.284, 5.483, 6.234, 6.938, 6.268, 5.562, 6.016) vnFrame <- data.frame(Melbourne=melData, Sydney=sydData) vn <- ts(vnFrame, start=c(1998, 1), frequency=4) str(vn)## Time-Series [1:72, 1:2] from 1998 to 2016: 4.87 4.11 4.42 5.17 5.55 ... ## - attr(*, "dimnames")=List of 2 ## ..$ : NULL ## ..$ : chr [1:2] "Melbourne" "Sydney"# Create three training series omitting the last 1, 2, and 3 years train1 <- window(vn[, "Melbourne"], end = c(2014, 4)) train2 <- window(vn[, "Melbourne"], end = c(2013, 4)) train3 <- window(vn[, "Melbourne"], end = c(2012, 4)) # Produce forecasts using snaive() fc1 <- snaive(train1, h = 4) fc2 <- snaive(train2, h = 4) fc3 <- snaive(train3, h = 4) # Use accuracy() to compare the MAPE of each series accuracy(fc1, vn[, "Melbourne"])["Test set", "MAPE"]## [1] 5.474755accuracy(fc2, vn[, "Melbourne"])["Test set", "MAPE"]## [1] 12.50411accuracy(fc3, vn[, "Melbourne"])["Test set", "MAPE"]## [1] 7.954534# Compute cross-validated errors for up to 8 steps ahead e <- matrix(NA_real_, nrow = 1000, ncol = 8) for (h in 1:8) e[, h] <- tsCV(goog, forecastfunction = naive, h = h) # Compute the MSE values and remove missing values mse <- colMeans(e^2, na.rm = TRUE) # Plot the MSE values against the forecast horizon data.frame(h = 1:8, MSE = mse) %>% ggplot(aes(x = h, y = MSE)) + geom_point()
Chapter 3 - Exponential smoothing
Exponentially weighted forecasts (simple exponential smoothing):
- The point estimates are represented as y-hat(i)
- The point estimate for y-hat(y) = alpha * y-hat(t-1) + alpha * (1 - alpha) * y-hat(t-2) + alpha * (1 - alpha)^2 * y-hat(t-3) + … where the extent of the looking back can be user-specified
- Smaller alpha leads to slower decay
- Smaller alpha leads to slower decay
- Can instead think of the forecast as y-hat(t) = alpha * y(t) + (1 - alpha) * y(t-1)
- Basically, need to estimate both alpha and the starting point
- Basically, need to estimate both alpha and the starting point
- Can run these models using myFC <- ses(myTS, h=)
- Behind the scenes, this is a non-linear optimization
- The h is the number of years for the forecast to project forwards
- Behind the scenes, this is a non-linear optimization
- While a simple method, this typically forms the starting point for future models
- Process for review for test-train analysis with time series data
- First, import and load your data. Determine how much of your data you want to allocate to training, and how much to testing; the sets should not overlap
- Subset the data to create a training set, which you will use as an argument in your forecasting function(s). Optionally, you can also create a test set to use later
- Compute forecasts of the training set using whichever forecasting function(s) you choose, and set h equal to the number of values you want to forecast, which is also the length of the test set
- To view the results, use the accuracy() function with the forecast as the first argument and original data (or test set) as the second
- Pick a measure in the output, such as RMSE or MAE, to evaluate the forecast(s); a smaller error indicates higher accuracy
- First, import and load your data. Determine how much of your data you want to allocate to training, and how much to testing; the sets should not overlap
Exponential smoothing methods with trend:
- Forecast is now level(t) + h * slope(t)
- Level is now alpha * y(t) + (1 - alpha) * (level(t-1) + slope(t-1))
- Trend (local linear trend) is allowed to change over time and is Beta * (level(t) - level(t-1)) + (1 - Beta) * slope(t-1)
- A smaller Beta means the slope changes just a little, while a higher Beta means the slope can change very quickly
- A smaller Beta means the slope changes just a little, while a higher Beta means the slope can change very quickly
- There are now four parameters to estimate - alpha and Beta, plus starting conditions level-0 and slope-0
- Parameters are again created to minimize SSE, using a methodology first developed by Holt
- myTS %>% holt(h=) %>% autoplot()
- Parameters are again created to minimize SSE, using a methodology first developed by Holt
- Alternately, can add a damping feature so that the data level off over time - Damped Trend Method
- There is an additional parameter 0 <= phi <= 1 where phi determines the damping (phi=1 is Holt)
- The damped=TRUE argument to holt() will find the parameter phi, in addition to the other Holt function outputs
- There is an additional parameter 0 <= phi <= 1 where phi determines the damping (phi=1 is Holt)
Exponential smoothing methods with trend and seasonality (commonly known as the Holt-Winters method):
- There is an additive version and a multiplicative version of the formula
- In the additive component, the seasoanl components average to zero
- In the multiplicative component, the seasoanl components average to one
- In the additive component, the seasoanl components average to zero
- Can run the models using hw(myTS, seasonal=) # seasonal can be “additive” or “multiplicative”
- Can think of a 3x3 grid of the potential forecast models
- Trend - N (None), A (Additive), Ad (Additive Damped)
- Seasonal - N (None), A (Additive), M (Multiplicative)
- Trend - N (None), A (Additive), Ad (Additive Damped)
- Model can be thought of as handling various portions of the grid, using(Trend, Seasonal) notation
- (N, N) - ses()
- (A, N) - holt()
- (Ad, N) - hw()
- (A, A) - hw()
- (A, M) - hw()
- (Ad, M) - hw()
- The other patterns, especially seasonality absent trend, will be considered later in the course
- (N, N) - ses()
- The Holt-Winters method can also be used for daily type of data, where the seasonal pattern is of length 7, and the appropriate unit of time for h is in days
State space models for exponential smoothing:
- There are 18 possible “innovations state space” models - Trend (3) x Seasonal (3) x Error (2)
- Error can be (A, M) for Additive or Multiplicative
- Multiplicative errors means that the error increases as the value increases (error tends to a fixed percentage rather than a fixed value)
- Error can be (A, M) for Additive or Multiplicative
- Overall, these are referred to as ETS models, for “Error, Trend, Seasonality”
- Parameters can be estimated using the likelihhod method, the probability of data arising from the specified model
- For additive errors, the goal is to minimize SSE
- Can choose the best model by minimizing a corrected version of Akaike’s Information Criteria (AICc), roughly the same as cross-validation but much faster
- Parameters can be estimated using the likelihhod method, the probability of data arising from the specified model
- By running ets(myTS), all of the work is done behind the scenes, and the function reports the best model based on minimizing AICc
- Maximizes likelihood rather than minimizing errors
- To produce a forecast, the model created by ets() must be piped to the forecast() function
- The biggest advantage of ets() is that the model is chosen for you
- Maximizes likelihood rather than minimizing errors
- The second argument for tsCV() must return a forecast object, so you need a function to fit a model and return forecasts. Recall:
- args(tsCV)
- function (y, forecastfunction, h = 1, …)
- args(tsCV)
Example code includes:
# You will also use summary() and fitted(), along with autolayer() for the first time, which is like autoplot() but it adds a "layer" to a plot rather than creating a new plot. # Here, you will apply these functions to marathon, the annual winning times in the Boston marathon from 1897-2016. The data are available in your workspace. # Use ses() to forecast the next 10 years of winning times data(marathon, package="fpp2") str(marathon)## Time-Series [1:120] from 1897 to 2016: 175 162 175 160 149 ...fc <- ses(marathon, h = 10) # Use summary() to see the model parameters summary(fc)## ## Forecast method: Simple exponential smoothing ## ## Model Information: ## Simple exponential smoothing ## ## Call: ## ses(y = marathon, h = 10) ## ## Smoothing parameters: ## alpha = 0.3457 ## ## Initial states: ## l = 167.1765 ## ## sigma: 5.4728 ## ## AIC AICc BIC ## 988.4474 988.6543 996.8099 ## ## Error measures: ## ME RMSE MAE MPE MAPE MASE ## Training set -0.8875951 5.472771 3.826287 -0.7098466 2.637645 0.8925669 ## ACF1 ## Training set -0.01207536 ## ## Forecasts: ## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 ## 2017 130.3562 123.3425 137.3698 119.6297 141.0826 ## 2018 130.3562 122.9353 137.7771 119.0069 141.7054 ## 2019 130.3562 122.5492 138.1631 118.4165 142.2958 ## 2020 130.3562 122.1814 138.5309 117.8539 142.8584 ## 2021 130.3562 121.8294 138.8829 117.3156 143.3967 ## 2022 130.3562 121.4914 139.2209 116.7987 143.9136 ## 2023 130.3562 121.1658 139.5465 116.3008 144.4116 ## 2024 130.3562 120.8514 139.8610 115.8199 144.8925 ## 2025 130.3562 120.5470 140.1653 115.3544 145.3580 ## 2026 130.3562 120.2518 140.4605 114.9029 145.8094# Use autoplot() to plot the forecasts autoplot(fc)
# Add the one-step forecasts for the training data to the plot autoplot(fc) + autolayer(fitted(fc))
# Create a training set using subset.ts() train <- subset(marathon, end = length(marathon) - 20) # Compute SES and naive forecasts, save to fcses and fcnaive fcses <- ses(train, h = 20) fcnaive <- naive(train, h = 20) # Calculate forecast accuracy measures accuracy(fcses, marathon)## ME RMSE MAE MPE MAPE MASE ## Training set -1.085512 5.863790 4.155943 -0.8606360 2.827999 0.8990895 ## Test set 0.457428 2.493965 1.894228 0.3171688 1.463856 0.4097941 ## ACF1 Theil's U ## Training set -0.01587645 NA ## Test set -0.12556096 0.6870722accuracy(fcnaive, marathon)## ME RMSE MAE MPE MAPE MASE ## Training set -0.4638047 6.904742 4.622391 -0.4086317 3.123559 1.0000000 ## Test set 0.2266667 2.462113 1.846667 0.1388780 1.429608 0.3995047 ## ACF1 Theil's U ## Training set -0.3589323 NA ## Test set -0.1255610 0.6799062# Save the best forecasts as fcbest # fcbest <- fcnaive # Here, you will apply it to the austa series, which contains annual counts of international visitors to Australia from 1980-2015 (in millions). The data has been pre-loaded into your workspace. # Produce 10 year forecasts of austa using holt() data(austa, package="fpp2") str(austa)## Time-Series [1:36] from 1980 to 2015: 0.83 0.86 0.877 0.867 0.932 ...fcholt <- holt(austa, h=10) # Look at fitted model using summary() summary(fcholt)## ## Forecast method: Holt's method ## ## Model Information: ## Holt's method ## ## Call: ## holt(y = austa, h = 10) ## ## Smoothing parameters: ## alpha = 0.9999 ## beta = 1e-04 ## ## Initial states: ## l = 0.5684 ## b = 0.1755 ## ## sigma: 0.1839 ## ## AIC AICc BIC ## 17.08684 19.08684 25.00443 ## ## Error measures: ## ME RMSE MAE MPE MAPE ## Training set -0.0006980015 0.1839059 0.1628927 -1.231661 6.322328 ## MASE ACF1 ## Training set 0.7994647 0.234277 ## ## Forecasts: ## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 ## 2016 7.034379 6.798694 7.270064 6.673930 7.394828 ## 2017 7.209838 6.876529 7.543147 6.700086 7.719590 ## 2018 7.385297 6.977065 7.793529 6.760960 8.009633 ## 2019 7.560756 7.089350 8.032161 6.839804 8.281707 ## 2020 7.736214 7.209144 8.263285 6.930129 8.542299 ## 2021 7.911673 7.334269 8.489077 7.028610 8.794736 ## 2022 8.087132 7.463435 8.710829 7.133269 9.040994 ## 2023 8.262591 7.595798 8.929383 7.242820 9.282362 ## 2024 8.438049 7.730775 9.145324 7.356366 9.519733 ## 2025 8.613508 7.867939 9.359078 7.473258 9.753758# Plot the forecasts autoplot(fcholt)
# Check that the residuals look like white noise checkresiduals(fcholt)
## ## Ljung-Box test ## ## data: Residuals from Holt's method ## Q* = 5.4561, df = 6, p-value = 0.4868 ## ## Model df: 4. Total lags used: 10# Here, you will apply hw() to a10, the monthly sales of anti-diabetic drugs in Australia from 1991 to 2008. The data are available in your workspace. # Plot the data data(a10, package="fpp2") str(a10)## Time-Series [1:204] from 1992 to 2008: 3.53 3.18 3.25 3.61 3.57 ...autoplot(a10)# Produce 3 year forecasts fc <- hw(a10, seasonal = "multiplicative", h = 3) # Check if residuals look like white noise checkresiduals(fc)
## ## Ljung-Box test ## ## data: Residuals from Holt-Winters' multiplicative method ## Q* = 55.57, df = 8, p-value = 3.421e-09 ## ## Model df: 16. Total lags used: 24whitenoise <- FALSE # Plot forecasts autoplot(fc)
# Here, you will compare an additive Holt-Winters method and a seasonal naive() method for the hyndsight data, which contains the daily pageviews on the Hyndsight blog for one year starting April 30, 2014 # Create training data with subset() train <- subset(hyndsight, end = length(hyndsight) - 28) # Holt-Winters additive forecasts as fchw fchw <- hw(train, seasonal = "additive", h = 28) # Seasonal naive forecasts as fcsn fcsn <- snaive(train, h=28) # Find better forecasts with accuracy() accuracy(fchw, hyndsight)## ME RMSE MAE MPE MAPE MASE ## Training set -0.06894788 230.0922 164.2099 -2.6329101 13.848691 0.7455151 ## Test set 41.12766744 180.6719 143.8484 0.9333033 9.254484 0.6530736 ## ACF1 Theil's U ## Training set 0.2145322 NA ## Test set 0.2337612 0.4233897accuracy(fcsn, hyndsight)## ME RMSE MAE MPE MAPE MASE ## Training set 10.50 310.3282 220.2636 -2.1239387 18.01077 1.0000000 ## Test set 0.25 202.7610 160.4643 -0.6888732 10.25880 0.7285101 ## ACF1 Theil's U ## Training set 0.4255730 NA ## Test set 0.3089795 0.450266# Plot the better forecasts autoplot(fchw)
# Fit ETS model to austa in fitaus fitaus <- ets(austa) # Check residuals checkresiduals(fitaus)
## ## Ljung-Box test ## ## data: Residuals from ETS(A,A,N) ## Q* = 5.4561, df = 6, p-value = 0.4868 ## ## Model df: 4. Total lags used: 10# Plot forecasts autoplot(forecast(fitaus))
# Repeat for hyndsight data in fiths fiths <- ets(hyndsight) checkresiduals(fiths)
## ## Ljung-Box test ## ## data: Residuals from ETS(A,N,A) ## Q* = 65.856, df = 5, p-value = 7.444e-13 ## ## Model df: 9. Total lags used: 14autoplot(forecast(fiths))
# Which model(s) fails test? (TRUE or FALSE) fitausfail <- FALSE fithsfail <- TRUE # Function to return ETS forecasts fets <- function(y, h) { forecast(ets(y), h = h) } data(qcement, package="fpp2") str(qcement)## Time-Series [1:233] from 1956 to 2014: 0.465 0.532 0.561 0.57 0.529 0.604 0.603 0.582 0.554 0.62 ...cement <- window(qcement, start=1994) str(cement)## Time-Series [1:81] from 1994 to 2014: 1.47 1.75 1.96 1.83 1.63 ...# Apply tsCV() for both methods e1 <- tsCV(cement, fets, h = 4) e2 <- tsCV(cement, snaive, h = 4) # Compute MSE of resulting errors (watch out for missing values) mean(e1^2, na.rm=TRUE)## [1] 0.04442133mean(e2^2, na.rm=TRUE)## [1] 0.02921384# Copy the best forecast MSE bestmse <- mean(e2^2, na.rm=TRUE) # Computing the ETS does not work well for all series # Here, you will observe why it does not work well for the annual Canadian lynx population available in your workspace as lynx # Plot the lynx series data(lynx) str(lynx)## Time-Series [1:114] from 1821 to 1934: 269 321 585 871 1475 ...autoplot(lynx)
# Use ets() to model the lynx series fit <- ets(lynx) # Use summary() to look at model and parameters summary(fit)## ETS(M,N,N) ## ## Call: ## ets(y = lynx) ## ## Smoothing parameters: ## alpha = 0.9999 ## ## Initial states: ## l = 169.2223 ## ## sigma: 0.9489 ## ## AIC AICc BIC ## 2052.369 2052.587 2060.578 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE MASE ## Training set 28.30726 1182.181 824.4878 -44.94341 94.83425 0.9923325 ## ACF1 ## Training set 0.3785704# Plot 20-year forecasts of the lynx series fit %>% forecast(h=20) %>% autoplot()
Chapter 4 - Forecasting with ARIMA Models
Transformations for variance stabilization:
- Common transformations include powers, logs, inverses, and the like
- There is a family of transformations for time series data known as the Box-Cox transformations
- There is a lambda paremeter
- lambda = 1 is no transformation
- lambda = 0.5 is like a square root
- lambda = 0 is like a natural logarithm
- lambda = -1 is like an inverse
- There is a lambda paremeter
- Can run BoxCox.lambda(myTS) to get a rough approximation of the best parameter to use for further analysis
- Can add the lambda= argument to the ets() function, and R will then take care of the rest
- The forecast() function gets the lambda and other parameters, and thereby puts the forecast back on the normal scale
- The forecast() function gets the lambda and other parameters, and thereby puts the forecast back on the normal scale
- The lambda is not so often used with ets() models since they can handle multiplicative error already, but it is vital for ARIMA models (next chapter)
- Differencing is a way of making a time series stationary; this means that you remove any systematic patterns such as trend and seasonality from the data
- A white noise series is considered a special case of a stationary time series
- With non-seasonal data, you use lag-1 differences to model changes between observations rather than the observations directly
- You have done this before by using the diff() function
- A white noise series is considered a special case of a stationary time series
- With seasonal data, differences are often taken between observations in the same season of consecutive years, rather than in consecutive periods
- For example, with quarterly data, one would take the difference between Q1 in one year and Q1 in the previous year
- This is called seasonal differencing
- Sometimes you need to apply both seasonal differences and lag-1 differences to the same series, thus, calculating the differences in the differences
- For example, with quarterly data, one would take the difference between Q1 in one year and Q1 in the previous year
ARIMA models - AutoRegressive Integrated Moving Average:
- The AR component is a multiple regression against the lagged observations of the model, using the last p observations
- The MA component is a multiple regression against the lagged errors of the model, using the last q errors
- Combining these, you have an ARMA(p, q) model
- ARMA models only work for stationary data, so it must have been differenced or otherwise transformed prior to running these
- ARMA models only work for stationary data, so it must have been differenced or otherwise transformed prior to running these
- The ARIMA(p, d, q) is an ARMA(p, q) model with d level differencing included also
- Can run auto.arima(myTS) to get R to pick what it believes are the best parameters for p, d, q
- The “drift” estimate means the intercept, c, for the AR or the MA portion of the model
- Can run auto.arima(myTS) to get R to pick what it believes are the best parameters for p, d, q
- Note that AICc can only be compared for models of the same class (including the same value for d if the model is ARIMA)
- Due to the search grid criteria, auto.arima can sometimes get a non-optimal solution, since the optimal solution was not in its search grid
- To make auto.arima() work harder to find a good model, add the optional argument stepwise = FALSE to look at a much larger collection of models
- To make auto.arima() work harder to find a good model, add the optional argument stepwise = FALSE to look at a much larger collection of models
- The Arima() function can be used to select a specific ARIMA model
- Its first argument, order, is set to a vector that specifies the values of pp, dd and qq
- The second argument, include.constant, is a booolean that determines if the constant cc, or drift, should be included
- Its first argument, order, is set to a vector that specifies the values of pp, dd and qq
- Below is an example of a pipe function that would plot forecasts of usnetelec from an ARIMA(2,1,2) model with drift:
- usnetelec %>% Arima(order = c(2,1,2), include.constant = TRUE) %>% forecast() %>% autoplot()
Seasonal ARIMA models - just needs a lot more differencing and lags:
- A seasonal ARIMA model is referred to as (p, d, q) with (P, D, Q)m
- The lower case letters are the main model
- The upper case letters are the seasonal component of the model, with m meaning the seasonal period
- ARIMA(0,1,4)(0,1,1)[12] # (p, d, q) = (0, 1, 4) and (P, D, Q) = (0, 1, 1) and m=12
- The lower case letters are the main model
- Prior to putting any data in to ARIMA, the variance issues need to have been solved; Box-Cox is the typical transformation
- The auto.arima() can handle all the other components
- Alternately, the auto.arima(lambda=) can be used
- The auto.arima() can handle all the other components
- The ARIMA models allow for seasoanlity to change, and the more recent seasons are weighted more heavily
- Because the differencing term is included, the model naturally has a trend even in the absence of a formal constant being included
- What happens when you want to create training and test sets for data that is more frequent than yearly?
- If needed, you can use a vector in form c(year, period) for the start and/or end keywords in the window() function
- You must also ensure that you’re using the appropriate values of h in forecasting functions
- Recall that h should be equal to the length of the data that makes up your test set
- If your data spans 15 years, your training set consists of the first 10 years, and you intend to forecast the last 5 years of data, you would use h = 12 * 5 not h = 5 because your test set would include 60 monthly observations
- If instead your training set consists of the first 9.5 years and you want forecast the last 5.5 years, you would use h = 66 to account for the extra 6 months
- If needed, you can use a vector in form c(year, period) for the start and/or end keywords in the window() function
Example code includes:
# Plot the series autoplot(a10)# Try four values of lambda in Box-Cox transformations a10 %>% BoxCox(lambda = 0.0) %>% autoplot()
a10 %>% BoxCox(lambda = 0.1) %>% autoplot()
a10 %>% BoxCox(lambda = 0.2) %>% autoplot()
a10 %>% BoxCox(lambda = 0.3) %>% autoplot()
# Compare with BoxCox.lambda() BoxCox.lambda(a10)## [1] 0.1313326# In this exercise, you will use the pre-loaded wmurders data, which contains the annual female murder rate in the US from 1950-2004 data(wmurders, package="fpp2") # Plot the US female murder rate autoplot(wmurders)
# Plot the differenced murder rate autoplot(diff(wmurders))
# Plot the ACF of the differenced murder rate ggAcf(diff(wmurders))
# In this exercise, you will use differencing and transformations simultaneously to make a time series look stationary. The data set here is h02, which contains 17 years of monthly corticosteroid drug sales in Australia data(h02, package="fpp2") str(h02)## Time-Series [1:204] from 1992 to 2008: 0.43 0.401 0.432 0.493 0.502 ...# Plot the data autoplot(h02)
# Take logs and seasonal differences of h02 difflogh02 <- diff(log(h02), lag = 12) # Plot difflogh02 autoplot(difflogh02)
# Take another difference and plot ddifflogh02 <- diff(difflogh02) autoplot(ddifflogh02)
# Plot ACF of ddifflogh02 ggAcf(ddifflogh02)
# Fit an automatic ARIMA model to the austa series fit <- auto.arima(austa) # Check that the residuals look like white noise checkresiduals(fit)
## ## Ljung-Box test ## ## data: Residuals from ARIMA(0,1,1) with drift ## Q* = 3.2552, df = 8, p-value = 0.9173 ## ## Model df: 2. Total lags used: 10residualsok <- TRUE # Summarize the model summary(fit)## Series: austa ## ARIMA(0,1,1) with drift ## ## Coefficients: ## ma1 drift ## 0.3006 0.1735 ## s.e. 0.1647 0.0390 ## ## sigma^2 estimated as 0.03376: log likelihood=10.62 ## AIC=-15.24 AICc=-14.46 BIC=-10.57 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE MASE ## Training set 0.0008313383 0.1759116 0.1520309 -1.069983 5.513269 0.7461559 ## ACF1 ## Training set -0.000571993# Find the AICc value and the number of differences used AICc <- round(fit$aicc, 2) d <- 1 # Plot forecasts of fit fit %>% forecast(h = 10) %>% autoplot()
# Plot forecasts from an ARIMA(0,1,1) model with no drift austa %>% Arima(order = c(0, 1, 1), include.constant = FALSE) %>% forecast() %>% autoplot()
# Plot forecasts from an ARIMA(2,1,3) model with drift austa %>% Arima(order = c(2, 1, 3), include.constant = TRUE) %>% forecast() %>% autoplot()
# Plot forecasts from an ARIMA(0,0,1) model with a constant austa %>% Arima(order = c(0, 0, 1), include.constant = TRUE) %>% forecast() %>% autoplot()
# Plot forecasts from an ARIMA(0,2,1) model with no constant austa %>% Arima(order = c(0, 2, 1), include.constant = FALSE) %>% forecast() %>% autoplot()
# Set up forecast functions for ETS and ARIMA models fets <- function(x, h) { forecast(ets(x), h = h) } farima <- function(x, h) { forecast(auto.arima(x), h=h) } # Compute CV errors for ETS as e1 e1 <- tsCV(austa, fets, h=1) # Compute CV errors for ARIMA as e2 e2 <- tsCV(austa, farima, h=1) # Find MSE of each model class mean(e1**2, na.rm=TRUE)## [1] 0.05684574mean(e2**2, na.rm=TRUE)## [1] 0.04336277# Plot 10-year forecasts using the best model class austa %>% farima(h=10) %>% autoplot()
# Check that the logged h02 data have stable variance h02 %>% log() %>% autoplot()
# Fit a seasonal ARIMA model to h02 with lambda = 0 fit <- auto.arima(h02, lambda=0) # Summarize the fitted model summary(fit)## Series: h02 ## ARIMA(2,1,3)(0,1,1)[12] ## Box Cox transformation: lambda= 0 ## ## Coefficients: ## ar1 ar2 ma1 ma2 ma3 sma1 ## -1.0194 -0.8351 0.1717 0.2578 -0.4206 -0.6528 ## s.e. 0.1648 0.1203 0.2079 0.1177 0.1060 0.0657 ## ## sigma^2 estimated as 0.004203: log likelihood=250.8 ## AIC=-487.6 AICc=-486.99 BIC=-464.83 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE ## Training set -0.003823286 0.05006017 0.03588577 -0.643286 4.52991 ## MASE ACF1 ## Training set 0.5919957 -0.007519928# Record the amount of lag-1 differencing and seasonal differencing used d <- 1 D <- 1 # Plot 2-year forecasts fit %>% forecast(h=24) %>% autoplot()
# Find an ARIMA model for euretail data(euretail, package="fpp2") str(euretail)## Time-Series [1:64] from 1996 to 2012: 89.1 89.5 89.9 90.1 89.2 ...fit1 <- auto.arima(euretail) # Don't use a stepwise search fit2 <- auto.arima(euretail, stepwise=FALSE) # AICc of better model AICc <- round(min(fit1$aicc, fit2$aicc), 2) # Compute 2-year forecasts from better model fit2 %>% forecast(h=8) %>% autoplot()
# In the final exercise for this chapter, you will compare seasonal ARIMA and ETS models applied to the quarterly cement production data qcement # Because the series is very long, you can afford to use a training and test set rather than time series cross-validation. This is much faster # Use 20 years of the qcement data beginning in 1988 train <- window(qcement, start = c(1988, 1), end = c(2007, 4)) # Fit an ARIMA and an ETS model to the training data fit1 <- auto.arima(train) fit2 <- ets(train) # Check that both models have white noise residuals checkresiduals(fit1)
## ## Ljung-Box test ## ## data: Residuals from ARIMA(2,0,0)(2,1,1)[4] with drift ## Q* = 3.5497, df = 3, p-value = 0.3144 ## ## Model df: 6. Total lags used: 9checkresiduals(fit2)
## ## Ljung-Box test ## ## data: Residuals from ETS(M,N,M) ## Q* = 7.1565, df = 3, p-value = 0.06707 ## ## Model df: 6. Total lags used: 9# Produce forecasts for each model fc1 <- forecast(fit1, h = length(window(qcement, start=2008))) fc2 <- forecast(fit2, h = length(window(qcement, start=2008))) # Use accuracy() to find better model based on RMSE accuracy(fc1, qcement)## ME RMSE MAE MPE MAPE ## Training set -0.005847264 0.1005547 0.07954692 -0.6550065 4.347836 ## Test set -0.158355693 0.1978197 0.16749405 -7.3004639 7.653446 ## MASE ACF1 Theil's U ## Training set 0.5434705 -0.01315073 NA ## Test set 1.1443319 0.28848376 0.7226779accuracy(fc2, qcement)## ME RMSE MAE MPE MAPE ## Training set 0.01497655 0.1017825 0.07904747 0.5666991 4.332964 ## Test set -0.13546124 0.1836477 0.15332875 -6.2632105 6.954592 ## MASE ACF1 Theil's U ## Training set 0.5400583 -0.03949197 NA ## Test set 1.0475535 0.54271907 0.6810648# bettermodel <- fc2
Chapter 5 - Advanced Methods
Dynamic Regression - could include factors like advertising or competition in a single model:
- In dynamic regression, the error term is an ARIMA series rather than a normal iid
- Example of forecasting personal consumption using personal income as one of the predictor variables
- Fitting a dynamic regression model is fairly straightforward with the auto.arima()
- fit <- auto.arima(myTS, xreg=myExplanatory) # note that xreg should be a matrix
- The forecast is then based on the predictors - forecast(fit, xreg=myPredictedExplanatory) # allows for scenario modeling
- fit <- auto.arima(myTS, xreg=myExplanatory) # note that xreg should be a matrix
Dynamic Harmonic Regression - handling periodic seasonality with Fourier terms:
- Fourier terms are of the form sin or cos of (2 * pi * k * t / m)
- m is the seasonal component
- k drives the harmonic frequencies
- Most types of seasonality can be modeled by the sin() and cos() pairing of the Fourier terms, properly tuned
- The error terms can then be assumed to be a non-seasonal ARIMA model
- m is the seasonal component
- One difference with the Fourier model is that it requires constant and un-changing seasonality
- Can run these models using auto.arima(myTS, xreg = fourier(myTS, K=), seasonal=FALSE, lambda=) # lambda to solve the variance issue, K to tune how deep the Fourier goes
- Can then do forecast(myModel, xreg=fourier(myTS, K=, h=)) # use the same K as for modeling, and use h as desired for time horizon periods (h= tells Fourier to look forward rather than backward)
- Can run these models using auto.arima(myTS, xreg = fourier(myTS, K=), seasonal=FALSE, lambda=) # lambda to solve the variance issue, K to tune how deep the Fourier goes
- Can add even more terms to the xreg; the main analyst duty is to test various K (never more than half the seasonal period) and pick the best (lowest) AICc
- The higher the order (K), the more “wiggly” the seasonal pattern is allowed to be
- With K=1, it is a simple sine curve
- You can select the value of KK by minimizing the AICc value
- The higher the order (K), the more “wiggly” the seasonal pattern is allowed to be
- With weekly data, it is difficult to handle seasonality using ETS or ARIMA models as the seasonal length is too large (approximately 52)
- Instead, you can use harmonic regression which uses sines and cosines to model the seasonality
- Instead, you can use harmonic regression which uses sines and cosines to model the seasonality
- Harmonic regressions are also useful when time series have multiple seasonal patterns
- For example, taylor contains half-hourly electricity demand in England and Wales over a few months in the year 2000
- The seasonal periods are 48 (daily seasonality) and 7 x 48 = 336 (weekly seasonality)
- For example, taylor contains half-hourly electricity demand in England and Wales over a few months in the year 2000
- auto.arima() would take a long time to fit a long time series such as this one, so instead you will fit a standard regression model with Fourier terms using the tslm() function
- This is very similar to lm() but is designed to handle time series
- With multiple seasonality, you need to specify the order KK for each of the seasonal periods
The formula argument is a symbolic description of the model to be fitted
args(tslm)
- function (formula, …)
- This is very similar to lm() but is designed to handle time series
TBATS models - combines many models in to a single model:
- Integrated model includes many core features
- Trigonometric terms for seasonality
- Box-Cox for heterogeneity
- ARMA for short-term dynamics
- Trend (possibly damped)
- Seasonal (including multiple and non-integer periods)
- Trigonometric terms for seasonality
- These models can be powerful but dangerous - sometimes, the automated choices are not so good
- Model will output TBATS(BoxCoxTerm, {p, q}, dampParam, {
- Model will output TBATS(BoxCoxTerm, {p, q}, dampParam, {
- The models are especially useful when the seasonal components are comples (many of them, interspersed with each other, etc.)
- Downside is model run time, and the potential for getting the wrong coefficients
Wrap up:
- Continual practice with methods to gain experience
- Book “Forecasting Principles and Practice” by Hyndman is available
- Book “Forecasting Principles and Practice” by Hyndman is available
- Try various techniques with various types of time series
Example code includes:
# In this exercise, you will model sales data regressed against advertising expenditure, with an ARMA error to account for any serial correlation in the regression errors # The data are available in your workspace as advert and comprise 24 months of sales and advertising expenditure for an automotive parts company data(advert, package="fma") str(advert)## Time-Series [1:24, 1:2] from 1 to 24: 25 0 15 10 20 10 5 5 15 15 ... ## - attr(*, "dimnames")=List of 2 ## ..$ : NULL ## ..$ : chr [1:2] "advert" "sales"# Time plot of both variables autoplot(advert, facets=TRUE)
# Fit ARIMA model fit <- auto.arima(advert[, "sales"], xreg = advert[, "advert"], stationary = TRUE) # Check model. Increase in sales for each unit increase in advertising salesincrease <- coefficients(fit)[3] # Forecast fit as fc fc <- forecast(fit, xreg = rep(10, 6)) # Plot fc with x and y labels autoplot(fc) + xlab("Month") + ylab("Sales")
data(elecdaily, package="fpp2") str(elecdaily)## Time-Series [1:365, 1:3] from 1 to 53: 175 189 189 174 170 ... ## - attr(*, "dimnames")=List of 2 ## ..$ : NULL ## ..$ : chr [1:3] "Demand" "WorkDay" "Temperature"elec <- elecdaily colnames(elec)[2] <- "Workday" # Time plots of demand and temperatures autoplot(elec[, c("Demand", "Temperature")], facets = TRUE)
# Matrix of regressors xreg <- cbind(MaxTemp = elec[, "Temperature"], MaxTempSq = elec[, "Temperature"] ** 2, Workday = elec[, "Workday"]) # Fit model fit <- auto.arima(elec[, "Demand"], xreg = xreg) # Forecast fit one day ahead forecast(fit, xreg = cbind(20, 20**2, 1))## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 ## 53.14286 185.4008 176.9271 193.8745 172.4414 198.3602# The pre-loaded gasoline data comprises weekly data on US finished motor gasoline products # In this exercise, you will fit a harmonic regression to this data set and forecast the next 3 years data(gasoline, package="fpp2") str(gasoline)## Time-Series [1:1355] from 1991 to 2017: 6621 6433 6582 7224 6875 ...# Set up harmonic regressors of order 13 harmonics <- fourier(gasoline, K = 13) # Fit regression model with ARIMA errors fit <- auto.arima(gasoline, xreg = harmonics, seasonal = FALSE) # Forecasts next 3 years newharmonics <- fourier(gasoline, K = 13, h = 156) fc <- forecast(fit, xreg = newharmonics) # Plot forecasts fc autoplot(fc)
# Fit a harmonic regression using order 10 for each type of seasonality fit <- tslm(taylor ~ fourier(taylor, K = c(10, 10))) # Forecast 20 working days ahead fc <- forecast(fit, newdata = data.frame(fourier(taylor, K = c(10, 10), h = 20 * 48))) # Plot the forecasts autoplot(fc)
# Check the residuals of fit checkresiduals(fit)
## ## Breusch-Godfrey test for serial correlation of order up to 672 ## ## data: Residuals from Linear regression model ## LM test = 3938.9, df = 672, p-value < 2.2e-16# Another time series with multiple seasonal periods is calls, which contains 20 consecutive days of 5-minute call volume data for a large North American bank # There are 169 5-minute periods in a working day, and so the weekly seasonal frequency is 5 x 169 = 845 # The weekly seasonality is relatively weak, so here you will just model daily seasonality. calls is pre-loaded into your workspace # The residuals in this case still fail the white noise tests, but their autocorrelations are tiny, even though they are significant # This is because the series is so long. It is often unrealistic to have residuals that pass the tests for such long series # The effect of the remaining correlations on the forecasts will be negligible data(calls, package="fpp2") calls <- window(calls, start=29.8) str(calls)## Time-Series [1:3380] from 29.8 to 33.8: 98 83 89 87 71 85 76 81 86 94 ... ## - attr(*, "names")= chr [1:3380] "X26.09.20031" "X26.09.20032" "X26.09.20033" "X26.09.20034" ... ## - attr(*, "msts")= num [1:2] 169 845# Plot the calls data autoplot(calls)
# Set up the xreg matrix xreg <- fourier(calls, K = c(10, 0)) # Fit a dynamic regression model fit <- auto.arima(calls, xreg = xreg, seasonal=FALSE, stationary=TRUE) # Check the residuals checkresiduals(fit)
## ## Ljung-Box test ## ## data: Residuals from Regression with ARIMA(3,0,1) errors ## Q* = 1843.9, df = 1665, p-value = 0.001318 ## ## Model df: 25. Total lags used: 1690# Plot forecasts for 10 working days ahead fc <- forecast(fit, xreg = fourier(calls, c(10, 0), h = 10 * 169)) autoplot(fc)
# The gas data contains Australian monthly gas production # A plot of the data shows the variance has changed a lot over time, so it needs a transformation # The seasonality has also changed shape over time, and there is a strong trend # This makes it an ideal series to test the tbats() function which is designed to handle these features # Plot the gas data autoplot(gas)# Fit a TBATS model to the gas data fit <- tbats(gas) # Forecast the series for the next 5 years fc <- forecast(fit, h=60) # Plot the forecasts autoplot(fc)
# Record the Box-Cox parameter and the order of the Fourier terms lambda <- round(as.vector(fc$model$lambda), 3) # 0.082 K <- fc$model$k.vector #5
Network Analysis in R
Chapter 1 - Introduction to Networks
What are social networks?
- Patterns of relationships can be represented as a graph (which can be a database representation or a visualization
- Vertex (node) refers to individuals, while edges refer to connections between individuals
- The graph data can be stored in two ways
- The adjacency matrix can be thought of as a matrix of all vertices (as rows and columns), with a 1 meaning edge-between, and a 0 meaning no edge
- The edge matrix is a two-column matrix containing each pair of vertices that have an edge between them
- The adjacency matrix can be thought of as a matrix of all vertices (as rows and columns), with a 1 meaning edge-between, and a 0 meaning no edge
- The igraph package in R can be used to manage graphs (social networks)
- g <- graph.edgelist(as.matrix(df), directed=FALSE) # creates graph object g with appropriate edges assuming df is a data-frame containing the edge data
- print(g) will return a lot of information - the start will be intA intB where intA=#vertices and intB=#edges
- g <- graph.edgelist(as.matrix(df), directed=FALSE) # creates graph object g with appropriate edges assuming df is a data-frame containing the edge data
- There are many functions available for extracting information from the graph
- V(g) # return all the vertices
- E(g) # return all the edges
- gorder(g) # number of vertices
- gsize(g) # number of edges
- plot(g) # visualization of the graph
- V(g) # return all the vertices
Network attributes - may want to add information about vertices and edges:
- Most common edge attribute is “weight” - plots as a thicker edge
- For example, frequency of flights or volume of messages or the like
- For example, frequency of flights or volume of messages or the like
- Can add data to existing vertices and edges
- Can use set_vertex_attr(myGraph, “myName”, myValues) and set_edge_attr(myGraph, “myName”, myValues) to add to an existing graph
- Can see the existing attributes with vertex_attr() or edge_attr()
- Can use set_vertex_attr(myGraph, “myName”, myValues) and set_edge_attr(myGraph, “myName”, myValues) to add to an existing graph
- Can create the metadata for the vertices and edges from the raw data
- graph_from_data_frame(d=myEdgesDF, vertices=myVerticesDF, directed=FALSE)
- graph_from_data_frame(d=myEdgesDF, vertices=myVerticesDF, directed=FALSE)
- Can subset the edges or vertices using the [[]] operators
- E(g)[[inc(“E)]] # will pull all edges that include E
- E(g)[[frequency >= 3]] # will pull all the edges where frequency attribute is 3+
- E(g)[[inc(“E)]] # will pull all edges that include E
- Can create vertex attributes using formula logic, for example, adding a “color” attribute to the vertices
- V(g)\(color <- ifelse( V(g)\)age > 22, “red”, “white” )
- plot(g, vertex.label.color = “black”)
- V(g)\(color <- ifelse( V(g)\)age > 22, “red”, “white” )
Network visualization principles - many options for creating and customizing the display:
- The best visualization provides immediate insight and information to the viewer
- Size, labels, colors, and shape are frequently adjusted as needed to best convey information
- Size is helpful for showing more central (important) vertices, as can labels (though risky, can be to much)
- Color and shape can be helpful for showing categorical data about vertices
- Edge line weights, colors, and shapes can help tease out the relative importance of various types of edges
- Size, labels, colors, and shape are frequently adjusted as needed to best convey information
- The igraph package includes all of the major ways to display - some basic rules include
- Minimize edge crossing
- Do not allow vertices to overlap
- Make edge lengths as uniform as possible
- Increase symmetry where possible
- Keep key vertices (nodes) closer to the center
- Minimize edge crossing
- Can add commands to plot(myGraph, layout=) # one example is layout.fruchterman.reingold(g)
- You can also stipulate the layout by providing a matrix of (x, y) coordinates for each vertex
- Here you use the layout_as_tree() function to generate the matrix m of coordinates
- Then pass m to the layout function in plot() to plot
- Here you use the layout_as_tree() function to generate the matrix m of coordinates
- Choosing a correct layout can be bewildering
- Fortunately igraph has a function layout_nicely() that tries to choose the most appropriate layout function for a given graph object
- Fortunately igraph has a function layout_nicely() that tries to choose the most appropriate layout function for a given graph object
- This is done by using delete_edges() which takes two arguments
- The first is the graph object and the second is the subset of edges to be removed
Example code includes:
# Load igraph library(igraph)## ## Attaching package: 'igraph'## The following object is masked from 'package:tidyr': ## ## crossing## The following objects are masked from 'package:lubridate': ## ## %--%, union## The following objects are masked from 'package:dplyr': ## ## as_data_frame, groups, union## The following objects are masked from 'package:stats': ## ## decompose, spectrum## The following object is masked from 'package:base': ## ## unionfriends <- readr::read_csv("./RInputFiles/friends.csv")## Parsed with column specification: ## cols( ## name1 = col_character(), ## name2 = col_character() ## )str(friends)## Classes 'tbl_df', 'tbl' and 'data.frame': 27 obs. of 2 variables: ## $ name1: chr "Jessie" "Jessie" "Sidney" "Sidney" ... ## $ name2: chr "Sidney" "Britt" "Britt" "Donnie" ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 2 ## .. ..$ name1: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## .. ..$ name2: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"# Inspect the first few rows of the dataframe 'friends' head(friends)## # A tibble: 6 x 2 ## name1 name2 ## <chr> <chr> ## 1 Jessie Sidney ## 2 Jessie Britt ## 3 Sidney Britt ## 4 Sidney Donnie ## 5 Karl Berry ## 6 Sidney Rene# Convert friends dataframe to a matrix friends.mat <- as.matrix(friends) # Convert friends matrix to an igraph object g <- graph.edgelist(friends.mat, directed = FALSE) # Make a very basic plot of the network plot(g)
# Subset vertices and edges V(g)## + 16/16 vertices, named, from ebe8820: ## [1] Jessie Sidney Britt Donnie Karl Berry Rene Shayne ## [9] Elisha Whitney Odell Lacy Eugene Jude Rickie TommyE(g)## + 27/27 edges from ebe8820 (vertex names): ## [1] Jessie --Sidney Jessie --Britt Sidney --Britt Sidney --Donnie ## [5] Karl --Berry Sidney --Rene Britt --Rene Sidney --Shayne ## [9] Sidney --Elisha Sidney --Whitney Jessie --Whitney Donnie --Odell ## [13] Sidney --Odell Rene --Whitney Donnie --Shayne Jessie --Lacy ## [17] Rene --Lacy Elisha --Eugene Eugene --Jude Berry --Odell ## [21] Odell --Rickie Karl --Odell Britt --Lacy Elisha --Jude ## [25] Whitney--Lacy Britt --Whitney Karl --Tommy# Count number of edges gsize(g)## [1] 27# Count number of vertices gorder(g)## [1] 16# Inspect the objects 'genders' and 'ages' genders <- c('M', 'F', 'F', 'M', 'M', 'M', 'F', 'M', 'M', 'F', 'M', 'F', 'M', 'F', 'M', 'M') ages <- c(18, 19, 21, 20, 22, 18, 23, 21, 22, 20, 20, 22, 21, 18, 19, 20) # Create new vertex attribute called 'gender' g <- set_vertex_attr(g, "gender", value = genders) # Create new vertex attribute called 'age' g <- set_vertex_attr(g, "age", value = ages) # View all vertex attributes in a list vertex_attr(g)## $name ## [1] "Jessie" "Sidney" "Britt" "Donnie" "Karl" "Berry" "Rene" ## [8] "Shayne" "Elisha" "Whitney" "Odell" "Lacy" "Eugene" "Jude" ## [15] "Rickie" "Tommy" ## ## $gender ## [1] "M" "F" "F" "M" "M" "M" "F" "M" "M" "F" "M" "F" "M" "F" "M" "M" ## ## $age ## [1] 18 19 21 20 22 18 23 21 22 20 20 22 21 18 19 20# View attributes of first five vertices in a dataframe V(g)[[1:5]]## + 5/16 vertices, named, from ebe8820: ## name gender age ## 1 Jessie M 18 ## 2 Sidney F 19 ## 3 Britt F 21 ## 4 Donnie M 20 ## 5 Karl M 22# View hours hours <- c(1, 2, 2, 1, 2, 5, 5, 1, 1, 3, 2, 1, 1, 5, 1, 2, 4, 1, 3, 1, 1, 1, 4, 1, 3, 3, 4) # Create new edge attribute called 'hours' g <- set_edge_attr(g, "hours", value = hours) # View edge attributes of graph object edge_attr(g)## $hours ## [1] 1 2 2 1 2 5 5 1 1 3 2 1 1 5 1 2 4 1 3 1 1 1 4 1 3 3 4# Find all edges that include "Britt" E(g)[[inc('Britt')]]## + 5/27 edges from ebe8820 (vertex names): ## tail head tid hid hours ## 2 Jessie Britt 1 3 2 ## 3 Sidney Britt 2 3 2 ## 7 Britt Rene 3 7 5 ## 23 Britt Lacy 3 12 4 ## 26 Britt Whitney 3 10 3# Find all pairs that spend 4 or more hours together per week E(g)[[hours>=4]]## + 6/27 edges from ebe8820 (vertex names): ## tail head tid hid hours ## 6 Sidney Rene 2 7 5 ## 7 Britt Rene 3 7 5 ## 14 Rene Whitney 7 10 5 ## 17 Rene Lacy 7 12 4 ## 23 Britt Lacy 3 12 4 ## 27 Karl Tommy 5 16 4friends1_nodes <- readr::read_csv("./RInputFiles/friends1_nodes.csv")## Parsed with column specification: ## cols( ## name = col_character(), ## gender = col_character() ## )str(friends1_nodes)## Classes 'tbl_df', 'tbl' and 'data.frame': 19 obs. of 2 variables: ## $ name : chr "Joe" "Erin" "Kelley" "Ronald" ... ## $ gender: chr "M" "F" "F" "M" ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 2 ## .. ..$ name : list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## .. ..$ gender: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"friends1_edges <- readr::read_csv("./RInputFiles/friends1_edges.csv")## Parsed with column specification: ## cols( ## name1 = col_character(), ## name2 = col_character(), ## hours = col_integer() ## )str(friends1_edges)## Classes 'tbl_df', 'tbl' and 'data.frame': 25 obs. of 3 variables: ## $ name1: chr "Joe" "Joe" "Joe" "Erin" ... ## $ name2: chr "Ronald" "Michael" "Troy" "Kelley" ... ## $ hours: int 1 3 2 3 5 1 3 5 2 1 ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 3 ## .. ..$ name1: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## .. ..$ name2: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## .. ..$ hours: list() ## .. .. ..- attr(*, "class")= chr "collector_integer" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"# Create an igraph object with attributes directly from dataframes g1 <- graph_from_data_frame(d = friends1_edges, vertices = friends1_nodes, directed = FALSE) # Subset edges greater than or equal to 5 hours E(g1)[[hours >= 5]]## + 4/25 edges from ec061c9 (vertex names): ## tail head tid hid hours ## 5 Kelley Valentine 3 6 5 ## 8 Ronald Jasmine 4 8 5 ## 12 Valentine Perry 6 15 5 ## 15 Jasmine Juan 8 9 6# Plot network and color vertices by gender V(g1)$color <- ifelse(V(g1)$gender == "F", "orange", "dodgerblue") plot(g1, vertex.label.color = "black")
# Plot the graph object g1 in a circle layout plot(g1, vertex.label.color = "black", layout = layout_in_circle(g1))
# Plot the graph object g1 in a Fruchterman-Reingold layout plot(g1, vertex.label.color = "black", layout = layout_with_fr(g1))
# Plot the graph object g1 in a Tree layout m <- layout_as_tree(g1) plot(g1, vertex.label.color = "black", layout = m)
# Plot the graph object g1 using igraph's chosen layout m1 <- layout_nicely(g1) plot(g1, vertex.label.color = "black", layout = m1)
# Create a vector of weights based on the number of hours each pair spend together w1 <- E(g1)$hours # Plot the network varying edges by weights m1 <- layout_nicely(g1) plot(g1, vertex.label.color = "black", edge.color = 'black', edge.width = w1, layout = m1)
# Create a new igraph object only including edges from the original graph that are greater than 2 hours long g2 <- delete_edges(g1, E(g1)[hours < 2]) # Plot the new graph w2 <- E(g2)$hours m2 <- layout_nicely(g2) plot(g2, vertex.label.color = "black", edge.color = 'black', edge.width = w2, layout = m2)
Chapter 2 - Identifying Important Vertices in a Network
Directed Networks - arrows represent the from-to relationship, such as e-mail exchanges:
- The first character after IGRAPH will be either “U” (undirected) or “D” (directed)
- Can also run is.directed(myGraph) and is.weighted(myGraph) to get the Boolean outcome
- Can also run is.directed(myGraph) and is.weighted(myGraph) to get the Boolean outcome
- The simplest measure of vertex influence is degree
- In undirected networks, vertices have degree equal to the total number of edges for that vertex
- In directed networks, vertices have in-degree equal to the total number of incoming edges and out-degree equal to the total number of outgoing edges
- In undirected networks, vertices have degree equal to the total number of edges for that vertex
- Can run several tests in R regarding social networks
- myGraph[“vert1”, “vert2”] will check whether an edge exists between these vertices
- If a 1 is returned that indicates TRUE there is an edge. If a 0 is returned that indicates FALSE there is not an edge
- incident(myGraph, “myVert”, mode=c(“all”)) will show all of the edges to or from “myVert”
- head_of(g, E(g)) # find the origin of all edges in the network
- myGraph[“vert1”, “vert2”] will check whether an edge exists between these vertices
Relationship between Vertices - overall patterns between networks (neighbors and paths):
- Can use neighbors as one measure of a vertices importance
- neighbors(myGraph, “myVertex”, mode=c(“all”)) # will list all of the neighbors for vertex “myVertex”
- If you have x as all the neighbors for x and y as all the neighbors for y, then intersection(x, y) will give any common neighbors of (one-stop paths between) x and y
- neighbors(myGraph, “myVertex”, mode=c(“all”)) # will list all of the neighbors for vertex “myVertex”
- Can consider the path-length as the number of stops needed to connect two points - 1 for neighbors, 2 for one-stop, 3 for two-stop, etc.
- The farthest_vertices(myGraph) call will find the longest-distance connection available in the graph
- The get_diameter(myGraph) will spell out the longest path
- Each of the above functions only returns one output, even if there are many tied for the longest path
- The ego(myGraph, maxDistance, “myVertex”, mode=c(“out”)) will return all vertices that can be reached in at most maxDistance using outbound edges only
- The farthest_vertices(myGraph) call will find the longest-distance connection available in the graph
- The inter-connectivity of a network can be assessed by examining the number and length of paths between vertices
- A path is simply the chain of connections between vertices
- The number of intervening edges between two vertices represents the geodesic distance between vertices
- Vertices that are connected to each other have a geodesic distance of 1
- Those that share a neighbor in common but are not connected to each other have a geodesic distance of 2 and so on
- In directed networks, the direction of edges can be taken into account
- If two vertices cannot be reached via following directed edges they are given a geodesic distance of infinity
- A path is simply the chain of connections between vertices
Significant nodes in a network:
- There are many potential measures of the importance of a vertex, including
- degree
- betweenness
- eigenvector centrality
- closeness centrality
- pagerank centrality
- degree
- As per previous, can calculate out-degree (edges out) and in-degree (edges in) for each vertex
- degree(myGraph, mode=c(“”)) # where mode can be “all”, “in”, or “out”, and the function returns a named vector
- degree(myGraph, mode=c(“”)) # where mode can be “all”, “in”, or “out”, and the function returns a named vector
- Can also calculate betweenness, the question of how often a given vertex lies on the shortest path through the network
- Higher betweenness means connecting many parts of the network
- Can calculate betweenness(myGraph, directed=, normalized= ) where directed is a boolean for whether it is a directed graph and normalized is a boolean for whether to normalize so they sum to 1
- Higher betweenness means connecting many parts of the network
Example code includes:
measles <- readr::read_csv("./RInputFiles/measles.csv")## Parsed with column specification: ## cols( ## from = col_integer(), ## to = col_integer() ## )str(measles)## Classes 'tbl_df', 'tbl' and 'data.frame': 184 obs. of 2 variables: ## $ from: int 45 45 172 180 45 180 42 45 182 45 ... ## $ to : int 1 2 3 4 5 6 7 8 9 10 ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 2 ## .. ..$ from: list() ## .. .. ..- attr(*, "class")= chr "collector_integer" "collector" ## .. ..$ to : list() ## .. .. ..- attr(*, "class")= chr "collector_integer" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"# Get the graph object g <- graph_from_data_frame(measles, directed = TRUE) # is the graph directed? is.directed(g)## [1] TRUE# Is the graph weighted? is.weighted(g)## [1] FALSE# Where does each edge originate from? table(head_of(g, E(g)))## ## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18 19 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 56 57 58 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 ## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## 184 185 186 187 ## 1 1 1 1# Make a basic plot plot(g, vertex.label.color = "black", edge.color = 'gray77', vertex.size = 0, edge.arrow.size = 0.1, layout = layout_nicely(g))
# Is there an edge going from vertex 184 to vertex 178? g['184', '178']## [1] 1# Is there an edge going from vertex 178 to vertex 184? g['178', '184']## [1] 0# Show all edges going to or from vertex 184 incident(g, '184', mode = c("all"))## + 6/184 edges from ecb34f1 (vertex names): ## [1] 184->45 184->182 184->181 184->178 184->183 184->177# Show all edges going out from vertex 184 incident(g, '184', mode = c("out"))## + 6/184 edges from ecb34f1 (vertex names): ## [1] 184->45 184->182 184->181 184->178 184->183 184->177# Identify all neighbors of vertex 12 regardless of direction neighbors(g, '12', mode = c('all'))## + 5/187 vertices, named, from ecb34f1: ## [1] 45 13 72 89 109# Identify other vertices that direct edges towards vertex 12 neighbors(g, '12', mode = c('in'))## + 1/187 vertex, named, from ecb34f1: ## [1] 45# Identify any vertices that receive an edge from vertex 42 and direct an edge to vertex 124 n1 <- neighbors(g, '42', mode = c('out')) n2 <- neighbors(g, '124', mode = c('in')) intersection(n1, n2)## + 1/187 vertex, named, from ecb34f1: ## [1] 7# Which two vertices are the furthest apart in the graph ? farthest_vertices(g)## $vertices ## + 2/187 vertices, named, from ecb34f1: ## [1] 184 162 ## ## $distance ## [1] 5# Shows the path sequence between two furthest apart vertices. get_diameter(g)## + 6/187 vertices, named, from ecb34f1: ## [1] 184 178 42 7 123 162# Identify vertices that are reachable within two connections from vertex 42 ego(g, 2, '42', mode = c('out'))## [[1]] ## + 13/187 vertices, named, from ecb34f1: ## [1] 42 7 106 43 123 101 120 124 125 128 129 108 127# Identify vertices that can reach vertex 42 within two connections ego(g, 2, '42', mode = c('in'))## [[1]] ## + 3/187 vertices, named, from ecb34f1: ## [1] 42 178 184# Calculate the out-degree of each vertex g.outd <- degree(g, mode = c("out")) # View a summary of out-degree table(g.outd)## g.outd ## 0 1 2 3 4 6 7 8 30 ## 125 21 16 12 6 2 3 1 1# Make a histogram of out-degrees hist(g.outd, breaks = 30)
# Find the vertex that has the maximum out-degree which.max(g.outd)## 45 ## 1# Calculate betweenness of each vertex g.b <- betweenness(g, directed = TRUE) # Show histogram of vertex betweenness hist(g.b, breaks = 80)
# Create plot with vertex size determined by betweenness score plot(g, vertex.label = NA, edge.color = 'black', vertex.size = sqrt(g.b)+1, edge.arrow.size = 0.05, layout = layout_nicely(g))
# Make an ego graph g184 <- make_ego_graph(g, diameter(g), nodes = '184', mode = c("all"))[[1]] # Get a vector of geodesic distances of all vertices from vertex 184 dists <- distances(g184, "184") # Create a color palette of length equal to the maximal geodesic distance plus one. colors <- c("black", "red", "orange", "blue", "dodgerblue", "cyan") # Set color attribute to vertices of network g184. V(g184)$color <- colors[dists+1] # Visualize the network based on geodesic distance from vertex 184 (patient zero). plot(g184, vertex.label = dists, vertex.label.color = "white", vertex.label.cex = .6, edge.color = 'black', vertex.size = 7, edge.arrow.size = .05, main = "Geodesic Distances from Patient Zero" )
Chapter 3 - Characterizing Network Structures
Introduction - Forrest Gump network dataset analysis:
- Each edge indicates that the characters were in a scene together (undirected network)
- Eigenvector centrality is the concept of 1) being connected to a lot of other vertices, and 2) being connected to vertices that are themselves connected to many other vertices
- eigen_centrality(myGraph)$vector # the $vector pulls the “vector” items from the list that is returned by eigen_centrality()
- eigen_centrality(myGraph)$vector # the $vector pulls the “vector” items from the list that is returned by eigen_centrality()
- The simplest measure of the overall structure of a network is its density - proportion of edges that exist relative to the maximum number of edges that could exist
- edge_density(myGraph)
- edge_density(myGraph)
- Can also calculate the average path length for a network
- mean_distance(myGraph, directed=)
Understanding network structures:
- Random graph technique is frequently used to better understand the structures of a network
- One of the simplest forms of the random graph is to have the same number of vertices and a similar density
- erdos.renyi.game(n=gorder(myGraph), p.or.m=edge_density(myGraph), type=“gnp”) will generate this
- This technique is especially helpful if you want to gauge whether a property of your original graph is unusual; how does it compare to 1,000 random graphs?
- One of the simplest forms of the random graph is to have the same number of vertices and a similar density
Network Substructures:
- Triangles (triads) in a network are of particular interest - looking at any group of three edges, they can be
- Closed - all linked to each other - can be found using triangles(myGraph)
- count_triangles(myGraph, vids=“myVertex”) # counts all triangles (closed) that include myVertex
- transitivity(myGraph, vids=“myVertex”, type=“local”) # closed triangles that include myVertex as a proportion of all triangles that could potentially include myVertex
- The denominator assumed in transitivity is n * (n-1) / 2 where n is the number of neighbors that myVertex has in myGraph
- Closed - all linked to each other - can be found using triangles(myGraph)
- Cliques are another important network sub-structure - in a clique, every vertex is connected to every other vertex
- All triangles made by a clique are closed
- The largest_cliques(myGraph) will return information about the largest clique
- max_cliques(myGraph) returns a list of cliques of each size
- All triangles made by a clique are closed
Example code includes:
# In this chapter you will use a social network based on the movie Forrest Gump # Each edge of the network indicates that those two characters were in at least one scene of the movie together # Therefore this network is undirected # To familiarize yourself with the network, you will first create the network object from the raw dataset # Then, you will identify key vertices using a measure called eigenvector centrality # Individuals with high eigenvector centrality are those that are highly connected to other highly connected individuals # You will then make an exploratory visualization of the network gump <- readr::read_csv("./RInputFiles/gump.csv")## Parsed with column specification: ## cols( ## V1 = col_character(), ## V2 = col_character() ## )str(gump)## Classes 'tbl_df', 'tbl' and 'data.frame': 271 obs. of 2 variables: ## $ V1: chr "ABBIE HOFFMAN" "ABBIE HOFFMAN" "ANCHORMAN" "ANCHORMAN" ... ## $ V2: chr "JENNY" "POLICEMAN" "FORREST" "LT DAN" ... ## - attr(*, "spec")=List of 2 ## ..$ cols :List of 2 ## .. ..$ V1: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## .. ..$ V2: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## ..$ default: list() ## .. ..- attr(*, "class")= chr "collector_guess" "collector" ## ..- attr(*, "class")= chr "col_spec"# Inspect Forrest Gump Movie dataset head(gump)## # A tibble: 6 x 2 ## V1 V2 ## <chr> <chr> ## 1 ABBIE HOFFMAN JENNY ## 2 ABBIE HOFFMAN POLICEMAN ## 3 ANCHORMAN FORREST ## 4 ANCHORMAN LT DAN ## 5 ANCHORMAN MARGO ## 6 ANCHORMAN MRS GUMP# Make an undirected network g <- graph_from_data_frame(gump, directed = FALSE) # Identify key nodes using eigenvector centrality g.ec <- eigen_centrality(g) which.max(g.ec$vector)## FORREST ## 36# Plot Forrest Gump Network plot(g, vertex.label.color = "black", vertex.label.cex = 0.6, vertex.size = 25*(g.ec$vector), edge.color = 'gray88', main = "Forrest Gump Network" )
# Get density of a graph gd <- edge_density(g) # Get the diameter of the graph g diameter(g, directed = FALSE)## [1] 4# Get the average path length of the graph g g.apl <- mean_distance(g, directed = FALSE) g.apl## [1] 1.994967# Create one random graph with the same number of nodes and edges as g g.random <- erdos.renyi.game(n = gorder(g), p.or.m = edge_density(g), type = "gnp") g.random## IGRAPH edb5f5c U--- 94 279 -- Erdos renyi (gnp) graph ## + attr: name (g/c), type (g/c), loops (g/l), p (g/n) ## + edges from edb5f5c: ## [1] 1-- 8 4-- 9 5--10 6--10 7--11 3--13 12--13 2--14 4--14 5--14 ## [11] 6--14 11--14 8--15 4--16 4--17 2--18 1--20 6--21 2--22 13--22 ## [21] 17--22 15--24 5--25 8--25 9--25 25--26 5--27 16--27 3--28 8--29 ## [31] 27--29 11--30 16--30 17--30 20--30 22--30 23--30 12--31 15--31 19--31 ## [41] 26--31 6--32 17--32 21--32 11--33 4--34 7--34 26--34 1--35 8--36 ## [51] 10--36 26--36 3--37 14--37 19--38 32--38 37--38 2--39 2--40 7--40 ## [61] 33--40 6--41 10--42 18--42 24--42 30--42 16--43 24--43 31--43 8--44 ## [71] 34--44 34--45 40--45 5--47 27--47 42--47 35--48 45--48 2--49 2--50 ## + ... omitted several edgesplot(g.random)
# Get density of new random graph `g.random` edge_density(g.random)## [1] 0.06382979#Get the average path length of the random graph g.random mean_distance(g.random, directed = FALSE)## [1] 2.702197g.apl=mean_distance(g) # Generate 1000 random graphs gl <- vector('list', 1000) for(i in 1:1000){ gl[[i]] <- erdos.renyi.game(n = gorder(g), p.or.m = gd, type = "gnp") } # Calculate average path length of 1000 random graphs gl.apl <- lapply(gl, FUN=mean_distance, directed = FALSE) gl.apls <- unlist(gl.apl) # Plot the distribution of average path lengths hist(gl.apls, xlim = range(c(1.5, 6))) abline(v = g.apl, col = "red", lty = 3, lwd=2)
# Calculate the proportion of graphs with an average path length lower than our observed sum(gl.apls < g.apl)/1000## [1] 0# Show all triangles in the network. matrix(triangles(g), nrow = 3)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 1 1 1 1 2 4 4 6 6 6 6 7 7 ## [3,] 83 38 39 66 68 57 24 27 75 40 45 8 69 ## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 8 11 11 11 12 12 13 14 14 14 14 ## [3,] 69 12 13 70 70 13 70 4 19 24 71 ## [,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 14 14 14 14 14 15 15 17 17 18 18 ## [3,] 65 57 62 63 64 21 72 22 42 5 28 ## [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 19 19 21 22 24 26 26 26 26 26 26 ## [3,] 71 63 72 42 57 73 52 47 48 49 50 ## [,47] [,48] [,49] [,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 27 27 27 28 28 30 30 30 34 38 38 ## [3,] 75 45 40 5 90 84 61 51 88 83 66 ## [,58] [,59] [,60] [,61] [,62] [,63] [,64] [,65] [,66] [,67] [,68] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 38 39 39 40 40 41 41 41 41 41 41 ## [3,] 39 83 66 75 45 1 3 6 7 8 11 ## [,69] [,70] [,71] [,72] [,73] [,74] [,75] [,76] [,77] [,78] [,79] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 41 41 41 41 41 41 41 41 41 41 41 ## [3,] 12 13 26 27 30 32 33 86 37 38 39 ## [,80] [,81] [,82] [,83] [,84] [,85] [,86] [,87] [,88] [,89] [,90] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 41 41 41 41 41 41 41 41 41 41 41 ## [3,] 40 43 44 45 47 48 49 50 51 52 53 ## [,91] [,92] [,93] [,94] [,95] [,96] [,97] [,98] [,99] [,100] [,101] ## [1,] 36 36 36 36 36 36 36 36 36 36 36 ## [2,] 41 41 41 41 41 41 41 41 41 41 41 ## [3,] 54 56 58 61 66 69 70 73 74 75 79 ## [,102] [,103] [,104] [,105] [,106] [,107] [,108] [,109] [,110] [,111] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 41 41 41 43 43 43 44 44 44 44 ## [3,] 82 83 84 82 54 53 2 3 9 14 ## [,112] [,113] [,114] [,115] [,116] [,117] [,118] [,119] [,120] [,121] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 44 44 44 44 44 44 44 44 44 44 ## [3,] 17 19 22 82 71 42 43 53 62 63 ## [,122] [,123] [,124] [,125] [,126] [,127] [,128] [,129] [,130] [,131] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 44 44 45 47 47 47 47 47 48 48 ## [3,] 64 65 75 73 52 50 48 49 73 52 ## [,132] [,133] [,134] [,135] [,136] [,137] [,138] [,139] [,140] [,141] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 48 48 49 49 49 50 50 51 51 52 ## [3,] 50 49 73 52 50 73 52 84 61 73 ## [,142] [,143] [,144] [,145] [,146] [,147] [,148] [,149] [,150] [,151] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 53 54 54 56 58 59 60 60 60 60 ## [3,] 82 87 56 89 79 92 2 20 23 25 ## [,152] [,153] [,154] [,155] [,156] [,157] [,158] [,159] [,160] [,161] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 60 60 60 61 62 62 62 62 63 64 ## [3,] 31 81 43 84 71 19 35 63 71 3 ## [,162] [,163] [,164] [,165] [,166] [,167] [,168] [,169] [,170] [,171] ## [1,] 36 36 36 36 36 36 36 36 36 36 ## [2,] 64 64 64 64 64 65 65 65 65 65 ## [3,] 71 19 63 62 46 4 71 19 24 64 ## [,172] [,173] [,174] [,175] [,176] [,177] [,178] [,179] [,180] [,181] ## [1,] 36 36 36 36 41 41 41 41 41 41 ## [2,] 65 65 65 66 1 1 1 1 6 6 ## [3,] 63 57 62 83 83 38 39 66 27 75 ## [,182] [,183] [,184] [,185] [,186] [,187] [,188] [,189] [,190] [,191] ## [1,] 41 41 41 41 41 41 41 41 41 41 ## [2,] 6 6 7 7 8 11 11 11 12 12 ## [3,] 40 45 8 69 69 12 13 70 70 13 ## [,192] [,193] [,194] [,195] [,196] [,197] [,198] [,199] [,200] [,201] ## [1,] 41 41 41 41 41 41 41 41 41 41 ## [2,] 13 26 26 26 26 26 26 27 27 27 ## [3,] 70 73 52 47 48 49 50 75 45 40 ## [,202] [,203] [,204] [,205] [,206] [,207] [,208] [,209] [,210] [,211] ## [1,] 41 41 41 41 41 41 41 41 41 41 ## [2,] 30 30 30 38 38 38 39 39 40 40 ## [3,] 84 61 51 83 66 39 83 66 75 45 ## [,212] [,213] [,214] [,215] [,216] [,217] [,218] [,219] [,220] [,221] ## [1,] 41 41 41 41 41 41 41 41 41 41 ## [2,] 43 43 43 44 44 44 44 45 47 47 ## [3,] 82 54 53 3 82 43 53 75 73 52 ## [,222] [,223] [,224] [,225] [,226] [,227] [,228] [,229] [,230] [,231] ## [1,] 41 41 41 41 41 41 41 41 41 41 ## [2,] 47 47 47 48 48 48 48 49 49 49 ## [3,] 50 48 49 73 52 50 49 73 52 50 ## [,232] [,233] [,234] [,235] [,236] [,237] [,238] [,239] [,240] [,241] ## [1,] 41 41 41 41 41 41 41 41 41 41 ## [2,] 50 50 51 51 52 53 54 58 58 61 ## [3,] 73 52 84 61 73 82 56 10 79 84 ## [,242] [,243] [,244] [,245] [,246] [,247] [,248] [,249] [,250] [,251] ## [1,] 41 44 44 44 44 44 44 44 44 44 ## [2,] 66 2 14 14 14 14 14 14 17 17 ## [3,] 83 67 19 71 65 62 63 64 22 42 ## [,252] [,253] [,254] [,255] [,256] [,257] [,258] [,259] [,260] [,261] ## [1,] 44 44 44 44 44 44 44 44 44 44 ## [2,] 19 19 22 43 43 53 62 62 62 63 ## [3,] 71 63 42 82 53 82 71 19 63 71 ## [,262] [,263] [,264] [,265] [,266] [,267] [,268] [,269] [,270] [,271] ## [1,] 44 44 44 44 44 44 44 44 44 44 ## [2,] 64 64 64 64 64 65 65 65 65 65 ## [3,] 3 71 19 63 62 71 19 64 63 62 ## [,272] [,273] [,274] [,275] [,276] [,277] [,278] [,279] [,280] [,281] ## [1,] 14 14 14 14 14 14 14 14 14 14 ## [2,] 4 4 19 19 24 65 65 65 65 65 ## [3,] 57 24 71 63 57 4 71 19 24 64 ## [,282] [,283] [,284] [,285] [,286] [,287] [,288] [,289] [,290] [,291] ## [1,] 14 14 14 14 14 14 14 14 14 14 ## [2,] 65 65 65 62 62 62 63 64 64 64 ## [3,] 63 57 62 71 19 63 71 71 19 63 ## [,292] [,293] [,294] [,295] [,296] [,297] [,298] [,299] [,300] [,301] ## [1,] 14 65 65 65 65 65 65 65 65 65 ## [2,] 64 4 4 19 19 24 64 64 64 64 ## [3,] 62 57 24 71 63 57 71 19 63 62 ## [,302] [,303] [,304] [,305] [,306] [,307] [,308] [,309] [,310] [,311] ## [1,] 65 65 65 65 64 64 64 64 64 64 ## [2,] 63 62 62 62 19 19 63 62 62 62 ## [3,] 71 71 19 63 71 63 71 71 19 63 ## [,312] [,313] [,314] [,315] [,316] [,317] [,318] [,319] [,320] [,321] ## [1,] 62 62 62 19 26 26 26 26 26 26 ## [2,] 19 19 63 63 52 47 47 47 47 47 ## [3,] 71 63 71 71 73 73 52 50 48 49 ## [,322] [,323] [,324] [,325] [,326] [,327] [,328] [,329] [,330] [,331] ## [1,] 26 26 26 26 26 26 26 26 26 47 ## [2,] 48 48 48 48 49 49 49 50 50 52 ## [3,] 73 52 50 49 73 52 50 73 52 73 ## [,332] [,333] [,334] [,335] [,336] [,337] [,338] [,339] [,340] [,341] ## [1,] 47 47 47 47 47 47 47 47 47 48 ## [2,] 50 50 48 48 48 48 49 49 49 52 ## [3,] 73 52 73 52 50 49 73 52 50 73 ## [,342] [,343] [,344] [,345] [,346] [,347] [,348] [,349] [,350] [,351] ## [1,] 48 48 48 48 48 49 49 49 50 43 ## [2,] 50 50 49 49 49 52 50 50 52 53 ## [3,] 73 52 73 52 50 73 73 52 73 82 ## [,352] [,353] [,354] [,355] [,356] [,357] [,358] [,359] [,360] [,361] ## [1,] 1 1 1 1 1 1 6 6 6 6 ## [2,] 38 38 38 39 39 66 27 27 27 40 ## [3,] 83 66 39 83 66 83 75 45 40 75 ## [,362] [,363] [,364] [,365] [,366] [,367] [,368] [,369] [,370] [,371] ## [1,] 6 6 27 27 27 38 38 38 39 40 ## [2,] 40 45 45 40 40 66 39 39 66 45 ## [3,] 45 75 75 75 45 83 83 66 83 75 ## [,372] [,373] [,374] [,375] [,376] [,377] [,378] [,379] [,380] [,381] ## [1,] 4 11 11 11 12 30 30 30 51 7 ## [2,] 24 12 12 13 13 61 51 51 61 8 ## [3,] 57 70 13 70 70 84 84 61 84 69 ## [,382] [,383] [,384] ## [1,] 17 18 15 ## [2,] 22 28 21 ## [3,] 42 5 72# Count the number of triangles that vertex "BUBBA" is in. count_triangles(g, vids='BUBBA')## [1] 37# Calculate the global transitivity of the network. g.tr <- transitivity(g) g.tr## [1] 0.1918082# Calculate the local transitivity for vertex BUBBA. transitivity(g, vids='BUBBA', type = "local")## [1] 0.6727273# One thousand random networks are stored in the list object gl g.tr <- 0.1918082 # Calculate the proportion of random graphs that have a transitivity higher than the transitivity of Forrest Gump's network, which you previously calculated and assigned to g.tr # Calculate average transitivity of 1000 random graphs gl.tr <- lapply(gl, FUN=transitivity) gl.trs <- unlist(gl.tr) # Get summary statistics of transitivity scores summary(gl.trs)## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.02488 0.05418 0.06119 0.06138 0.06835 0.09910# Calculate the proportion of graphs with a transitivity score higher than Forrest Gump's network. sum(gl.trs > g.tr)/1000## [1] 0# Identify the largest cliques in the network largest_cliques(g)## [[1]] ## + 9/94 vertices, named, from ed8ebe2: ## [1] FORREST STRONGARM BUBBA DALLAS LT DAN MAN SGT SIMS ## [8] SOLDIER SONG ## ## [[2]] ## + 9/94 vertices, named, from ed8ebe2: ## [1] FORREST JENNY EMCEE MAN # MAN #1 MAN #2 MAN #3 MAN #5 MEN# Determine all maximal cliques in the network and assign to object 'clq' clq <- max_cliques(g) # Calculate the size of each maximal clique. table(unlist(lapply(clq, length)))## ## 2 3 4 5 6 7 9 ## 12 24 7 2 4 2 2# Often in network visualization you will need to subset part of a network to inspect the inter-connections of particular vertices # Here, you will create a visualization of the largest cliques in the Forrest Gump network # In the last exercise you determined that there were two cliques of size 9 # You will plot these side-by-side after creating two new igraph objects by subsetting out these cliques from the main network # The function subgraph() enables you to choose which vertices to keep in a new network object # Assign largest cliques output to object 'lc' lc <- largest_cliques(g) # Create two new undirected subgraphs, each containing only the vertices of each largest clique. gs1 <- as.undirected(induced_subgraph(g, lc[[1]])) gs2 <- as.undirected(induced_subgraph(g, lc[[2]])) # Plot the two largest cliques side-by-side par(mfrow=c(1, 2)) # To plot two plots side-by-side plot(gs1, vertex.label.color = "black", vertex.label.cex = 0.9, vertex.size = 0, edge.color = 'gray28', main = "Largest Clique 1", layout = layout.circle(gs1) ) plot(gs2, vertex.label.color = "black", vertex.label.cex = 0.9, vertex.size = 0, edge.color = 'gray28', main = "Largest Clique 2", layout = layout.circle(gs2) )
par(mfrow=c(1, 1)) # To return to the defaults
Chapter 4 - Identifying Special Relationships
Close relationships: assortativity and reciprocity:
- Question is often whether vertices randomly associate, or whether they preference similar vertices - “do birds of a feather flock together”
- Assortativity is the preferential attachment of vertices to other vertices that are similar in numeric or other attributes
- assortativity(myGraph, myVertexAttaributeVector) will pull the overall assortativity - need to convert to factor so it is a numeric
- The value return will be +1 (only attach to similar) through 0 (no pattern) through -1 (only attach to dissimilar)
- assortativity(myGraph, myVertexAttaributeVector) will pull the overall assortativity - need to convert to factor so it is a numeric
- Can also look at whether vertices with a high degree preferentially attach to other vertices with a high degree
- assortativity.degree(myGraph, directed=) # will return a value from -1 to +1
- assortativity.degree(myGraph, directed=) # will return a value from -1 to +1
- Reciprocity is a measure of a directed network, specifically the number of outbound edges that share an inbound edge
- reciprocity(myGraph)
- Each direction counts as an edge, so if there are 12 two-way edges (6 two-way connections) and 8 one-way edges then reciprocity is 0.6
- reciprocity(myGraph)
Community detection - building on cliques, reciprocity, assortativity:
- General philosophy is similar to k-means - build “communities” where the connections within the community are much stronger than the connections outside the community
- Modules, groups, clusters, and communities all refer to the same concept when looking at networks and graphs
- The modularity score is used, and is a measure of similarity within to similarity without
- Can run fastgreedy.community(myGraph) # will get a baseline of communities building from ground-level
- Can run edge.betweenness.community(myGraph) # will get a baseline of communities splitting from full network
- Modules, groups, clusters, and communities all refer to the same concept when looking at networks and graphs
- Assuming a community object x exists, plot(x, myGraph) will plot the community as part of the graph
Interactive network visualizations:
- There are many R packages for visualizing networks, each with strengths and weaknesses
- The threejs library is especially good for visualizing large networks - integrates well with igraph
- threejs::graphjs(myGraph) # will graph the myGraph object
- If the vertex has a “color” or “value” attribute, that will be picked up automatically
- The network becomes fully interactive
- threejs::graphjs(myGraph) # will graph the myGraph object
Example code includes:
# Plot the network plot(g1)
# Convert the gender attribute into a numeric value values <- as.numeric(factor(V(g1)$gender)) # Calculate the assortativity of the network based on gender assortativity(g1, values)## [1] 0.1319444# Calculate the assortativity degree of the network assortativity.degree(g1, directed = FALSE)## [1] 0.4615385# Calculate the observed assortativity observed.assortativity <- assortativity(g1, values) # Calculate the assortativity of the network randomizing the gender attribute 1000 times results <- vector('list', 1000) for(i in 1:1000){ results[[i]] <- assortativity(g1, sample(values)) } # Plot the distribution of assortativity values and add a red vertical line at the original observed value hist(unlist(results)) abline(v = observed.assortativity, col = "red", lty = 3, lwd=2)
fromData <- c(1, 6, 8, 9, 11, 12, 3, 5, 7, 12, 2, 4, 5, 10, 13, 14, 3, 6, 9, 10, 13, 3, 8, 15, 1, 3, 9, 8, 9, 13, 5, 12, 13, 14, 1, 2, 3, 4, 7, 11, 12, 14, 3, 6, 14, 15, 4, 6, 9, 12, 3, 7, 8, 12, 14, 3, 4, 10, 11) toData <- c(15, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 15, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 15, 13, 13, 13, 13, 13, 14, 14, 14, 14) g <- graph_from_data_frame(data.frame(from=fromData, to=toData)) # Make a plot of the chimp grooming network plot(g, edge.color = "black", edge.arrow.size = 0.3, edge.arrow.width = 0.5)
# Calculate the reciprocity of the graph reciprocity(g)## [1] 0.2711864# The first community detection method you will try is fast-greedy community detection # You will use the Zachary Karate Club network # This social network contains 34 club members and 78 edges # Each edge indicates that those two club members interacted outside the karate club as well as at the club # Perform fast-greedy community detection on network graph fromData <- c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 9, 9, 9, 10, 14, 15, 15, 16, 16, 19, 19, 20, 21, 21, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33) toData <- c(2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 18, 20, 22, 32, 3, 4, 8, 14, 18, 20, 22, 31, 4, 8, 28, 29, 33, 10, 9, 14, 8, 13, 14, 7, 11, 7, 11, 17, 17, 31, 33, 34, 34, 34, 33, 34, 33, 34, 33, 34, 34, 33, 34, 33, 34, 26, 28, 33, 34, 30, 26, 28, 32, 32, 30, 34, 34, 32, 34, 33, 34, 33, 34, 33, 34, 34) g <- graph_from_data_frame(data.frame(from=fromData, to=toData), directed=FALSE) kc = fastgreedy.community(g) # Determine sizes of each community sizes(kc)## Community sizes ## 1 2 3 ## 8 17 9# Determine which individuals belong to which community membership(kc)## 1 2 3 4 5 6 7 9 10 14 15 16 19 20 21 23 24 25 26 27 28 29 30 31 32 ## 1 3 3 3 1 1 1 2 3 3 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 ## 33 8 11 12 13 18 22 17 34 ## 2 3 1 1 3 3 3 1 2# Plot the community structure of the network plot(kc, g)
# Perform edge-betweenness community detection on network graph gc = edge.betweenness.community(g) # Determine sizes of each community sizes(gc)## Community sizes ## 1 2 3 4 5 ## 10 6 5 12 1# Plot community networks determined by fast-greedy and edge-betweenness methods side-by-side par(mfrow = c(1, 2)) plot(kc, g) plot(gc, g)
par(mfrow = c(1, 1)) # In this course you have exclusively used igraph to make basic static network plots # There are many packages available to make network plots # One very useful one is threejs which allows you to make interactive network visualizations # This package also integrates seamlessly with igraph # In this exercise you will make a basic interactive network plot of the karate club network using the threejs package # Once you have produced the visualization be sure to move the network around with your mouse # You should be able to scroll in and out of the network as well as rotate the network library(igraph) library(threejs) # Set a vertex attribute called 'color' to 'dodgerblue' g <- set_vertex_attr(g, "color", value = "dodgerblue") # Redraw the graph and make the vertex size 1 (ActiveX does not work with browser) # graphjs(g, vertex.size = 1) # Create numerical vector of vertex eigenvector centralities ec <- as.numeric(eigen_centrality(g)$vector) # Create new vector 'v' that is equal to the square-root of 'ec' multiplied by 5 v <- 5*sqrt(ec) # Plot threejs plot of graph setting vertex size to v (ActiveX does not work with browser) # graphjs(g, vertex.size = v) # Create an object 'i' containin the memberships of the fast-greedy community detection i <- membership(kc) # Check the number of different communities sizes(kc)## Community sizes ## 1 2 3 ## 8 17 9# Add a color attribute to each vertex, setting the vertex color based on community membership g <- set_vertex_attr(g, "color", value = c("yellow", "blue", "red")[i]) # Plot the graph using threejs (ActiveX does not work with browser) # graphjs(g)
Spatial Statistics in R
Chapter 1 - Introduction
Problems in spatial statistics:
- Epidemics, susceptibility, locations, etc.
- Divisions of a lager area - healt disricts, counties, etc.
- Geostatistical data is the availability of data that has a spatial component
- At school we were taught to make the most of a piece of graph paper by scaling our data to fit the page
- R will usually follow this advice by making a plot fill the graphics window
- R will usually follow this advice by making a plot fill the graphics window
- With spatial data, this can cause misleading distortion that changes the distance and direction between pairs of points
- The data in the previous exercise was created in a tall, skinny rectangle, and it should always be shown as a tall, skinny rectangle
- If R stretches this to fill a wide graphics window then it is misrepresenting the relationship between events in the up-down and left-right directions
- The data in the previous exercise was created in a tall, skinny rectangle, and it should always be shown as a tall, skinny rectangle
- So spatial plots should have scales so that one unit in the X axis is the same size as one unit on the Y axis
- Circles will appear as circles and not ellipses, and squares will appear square
- The ratio of the Y axis scale to the X axis scale is called the aspect ratio of the plot
- Spatial data should always be presented with an aspect ratio of 1:1
- Circles will appear as circles and not ellipses, and squares will appear square
Simulation and testing with spatstat:
- A “point” is defined to be any specific (x, y) location on the 2D plane
- An “event” is a key data point; in the literature, a point is just a location while an event is an observation or a specific point of interest
- The “window” is the defined study area, and events outside the window are unobserved
- A “spatial point pattern” is a set of events inside a defined window
- A “spatial point process” is a stochastic process (RNG) for events inside a defined window
- The spatstat library stores spatial objects inside the ppp library
- Coordinates, window, marks, etc.
- Defaults to a unit square for the window
- Can plot(), print(), summary(), etc.
- Coordinates, window, marks, etc.
- The most basic type of spatial plot is “complete spatial randomness” (csr), where no part of the window differs from the others
- In the quadrat test, the window is sub-divided in to squares, and counts are taken within each of the squares
- The expected distribution would be the Poisson distribution, and the assessment of fit can be made using Chi-squared for counts by bucket
- Quadrat count tests are implemented using quadrat.test(), which takes a planar point pattern, ppp() object
- In the quadrat test, the window is sub-divided in to squares, and counts are taken within each of the squares
- A Poisson point process creates events according to a Poisson distribution with an intensity parameter specifying the expected events per unit area
- The total number of events generated is a single number from a Poisson distribution, so multiple realisations of the same process can easily have different numbers of events
Further testing:
- The quadrat test depends on selecting the right sub-window size, otherwise the test can lose power due to homogeneity
- One alternative test is the “nearest neighbors” test - find the “nearest neighbor” for each event, calculate the distance, and plot the distribution
- Can compare the ecdf with theoretical, accounting for edge-effect adjustments to theoretical, since events near the edge are deprived of potential close neighbors outside the window
- Can compare the ecdf with theoretical, accounting for edge-effect adjustments to theoretical, since events near the edge are deprived of potential close neighbors outside the window
- Another alternative test is the Ripley’s K Function
- Count the number of events within a circle of diameter d from a specificed event
- Plot the resulting function and compare with theoretical (~ pi * d**2)
- Can calculate p-values based on comparisons to random simulation
- If the observed data are greater than random CI at any specific value for d, that suggests clustering at around that distance
- Count the number of events within a circle of diameter d from a specificed event
- Spatial statistics frequently uses Monte Carlo simulation to calculate CI and evaluate hypothesis tests
Example code includes:
# The number of points to create n <- 200 # Set the range xmin <- 0 xmax <- 1 ymin <- 0 ymax <- 2 # Sample from a Uniform distribution x <- runif(n, xmin, xmax) y <- runif(n, ymin, ymax) # The ratio of the Y axis scale to the X axis scale is called the aspect ratio of the plot. Spatial data should always be presented with an aspect ratio of 1:1. # See pre-defined variables # ls.str() # Plot points and a rectangle mapxy <- function(a = NA){ plot(x, y, asp = a) rect(xmin, ymin, xmax, ymax) } mapxy(1)
# How do we create a uniform density point pattern in a circle? # We might first try selecting radius and angle uniformly. But that produces a "cluster" at small distances # Instead we sample the radius from a non-uniform distribution that scales linearly with distance, so we have fewer points at small radii and more at large radii # This exercise uses spatstat's disc() function, that creates a circular window. # Load the spatstat package library(spatstat)## Loading required package: spatstat.data## Loading required package: nlme## ## Attaching package: 'nlme'## The following object is masked from 'package:forecast': ## ## getResponse## The following object is masked from 'package:directlabels': ## ## gapply## The following object is masked from 'package:dplyr': ## ## collapse## Loading required package: rpart## ## spatstat 1.55-0 (nickname: 'Stunned Mullet') ## For an introduction to spatstat, type 'beginner'## ## Note: R version 3.3.3 (2017-03-06) is more than 9 months old; we strongly recommend upgrading to the latest version## ## Attaching package: 'spatstat'## The following object is masked from 'package:threejs': ## ## vertices## The following objects are masked from 'package:igraph': ## ## diameter, edges, is.connected, vertices## The following object is masked from 'package:lattice': ## ## panel.histogram# Create this many points, in a circle of this radius n_points <- 300 radius <- 10 # Generate uniform random numbers up to radius-squared r_squared <- runif(n_points, 0, radius**2) angle <- runif(n_points, 0, 2*pi) # Take the square root of the values to get a uniform spatial distribution x <- sqrt(r_squared) * cos(angle) y <- sqrt(r_squared) * sin(angle) plot(spatstat::disc(radius)) points(x, y)
# Some variables have been pre-defined # ls.str() # Set coordinates and window ppxy <- ppp(x = x, y = y, window = disc(radius)) # Test the point pattern qt <- quadrat.test(ppxy)## Warning: Some expected counts are small; chi^2 approximation may be ## inaccurate# Inspect the results plot(qt)
print(qt)## ## Chi-squared test of CSR using quadrat counts ## Pearson X2 statistic ## ## data: ppxy ## X2 = 26.268, df = 24, p-value = 0.6794 ## alternative hypothesis: two.sided ## ## Quadrats: 25 tiles (irregular windows)# In the previous exercise you used a set of 300 events scattered uniformly within a circle # If you repeated the generation of the events again you will still have 300 of them, but in different locations # The dataset of exactly 300 points is from a Poisson point process conditioned on the total being 300 # The spatstat package can generate Poisson spatial processes with the rpoispp() function given an intensity and a window, that are not conditioned on the total # Just as the random number generator functions in R start with an "r", most of the random point-pattern functions in spatstat start with an "r". # The area() function of spatstat will compute the area of a window such as a disc # Create a disc of radius 10 disc10 <- disc(10) # Compute the rate as count divided by area lambda <- 500 / area(disc10) # Create a point pattern object ppois <- rpoispp(lambda = lambda, win = disc10) # Plot the Poisson point pattern plot(ppois)
# The spatstat package also has functions for generating point patterns from other process modelsparameters. # These generally fall into one of two classes: clustered processes, where points occur together more than under a uniform Poisson process, # and regular (aka inhibitory) processes where points are more spaced apart than under a uniform intensity Poisson process # Some process models can generate patterns on a continuum from clustered through uniform to regular depending on their parameters # The quadrat.test() function can test against clustered or regular alternative hypotheses # By default it tests against either of those, but this can be changed with the alternative parameter to create a one-sided test. # A Thomas process is a clustered pattern where a number of "parent" points, uniformly distributed, create a number of "child" points in their neighborhood # The child points themselves form the pattern. This is an attractive point pattern, and makes sense for modelling things like trees, since new trees will grow near the original tree # Random Thomas point patterns can be generated using rThomas() # This takes three numbers that determine the intensity and clustering of the points, and a window object. # Conversely the points of a Strauss process cause a lowering in the probability of finding another point nearby # The parameters of a Strauss process can be such that it is a "hard-core" process, where no two points can be closer than a set threshold # Creating points from this process involves some clever simulation algorithms # This is a repulsive point pattern, and makes sense for modelling things like territorial animals, since the other animals of that species will avoid the territory of a given animal # Random Strauss point patterns can be generated using rStrauss() # This takes three numbers that determine the intensity and "territory" of the points, and a window object # Points generated by a Strauss process are sometimes called regularly spaced. # Create a disc of radius 10 disc10 <- disc(10) # Generate clustered points from a Thomas process set.seed(123) p_cluster <- rThomas(kappa = 0.35, scale = 1, mu = 3, win = disc10) plot(p_cluster)
# Run a quadrat test quadrat.test(p_cluster, alternative = "clustered")## Warning: Some expected counts are small; chi^2 approximation may be ## inaccurate## ## Chi-squared test of CSR using quadrat counts ## Pearson X2 statistic ## ## data: p_cluster ## X2 = 53.387, df = 24, p-value = 0.0005142 ## alternative hypothesis: clustered ## ## Quadrats: 25 tiles (irregular windows)# Regular points from a Strauss process set.seed(123) p_regular <- rStrauss(beta = 2.9, gamma = 0.025, R = .5, W = disc10)## Warning: Simulation will be performed in the containing rectangle and ## clipped to the original window.plot(p_regular)
# Run a quadrat test quadrat.test(p_regular, alternative = "regular")## Warning: Some expected counts are small; chi^2 approximation may be ## inaccurate## ## Chi-squared test of CSR using quadrat counts ## Pearson X2 statistic ## ## data: p_regular ## X2 = 8.57, df = 24, p-value = 0.001619 ## alternative hypothesis: regular ## ## Quadrats: 25 tiles (irregular windows)# Another way of assessing clustering and regularity is to consider each point, and how it relates to the other points # One simple measure is the distribution of the distances from each point to its nearest neighbor # The nndist() function in spatstat takes a point pattern and for each point returns the distance to its nearest neighbor # Instead of working with the nearest-neighbor density, as seen in the histogram, it can be easier to work with the cumulative distribution function, G(r) # This is the probability of a point having a nearest neighbour within a distance r # For a uniform Poisson process, G can be computed theoretically, and is G(r) = 1 - exp( - lambda * pi * r ^ 2) # You can compute G empirically from your data using Gest() and so compare with the theoretical value. # Events near the edge of the window might have had a nearest neighbor outside the window, and so unobserved # This will make the distance to its observed nearest neighbor larger than expected, biasing the estimate of G # There are several methods for correcting this bias # Plotting the output from Gest shows the theoretical cumulative distribution and several estimates of the cumulative distribution using different edge corrections # Often these edge corrections are almost indistinguishable, and the lines overlap # The plot can be used as a quick exploratory test of complete spatial randomness # Two ppp objects, p_poisson, and p_regular are defined for you # Point patterns are pre-defined p_poisson <- ppois p_poisson## Planar point pattern: 501 points ## window: polygonal boundary ## enclosing rectangle: [-10, 10] x [-10, 10] unitsp_regular## Planar point pattern: 325 points ## window: polygonal boundary ## enclosing rectangle: [-10, 10] x [-10, 10] units# Calc nearest-neighbor distances for Poisson point data nnd_poisson <- nndist(p_poisson) # Draw a histogram of nearest-neighbor distances hist(nnd_poisson)
# Estimate G(r) G_poisson <- Gest(p_poisson) # Plot G(r) vs. r plot(G_poisson)
# Repeat for regular point data nnd_regular <- nndist(p_regular) hist(nnd_regular)
G_regular <- Gest(p_regular) plot(G_regular)
# A number of other functions of point patterns have been developed # They are conventionally denoted by various capital letters, including F, H, J, K and L # The K-function is defined as the expected number of points within a distance of a point of the process, scaled by the intensity # Like G, this can be computed theoretically for a uniform Poisson process and is K(r) = pi * r ^ 2 - the area of a circle of that radius # Deviation from pi * r ^ 2 can indicate clustering or point inhibition # Computational estimates of K(r) are done using the Kest() function. # As with G calculations, K-function calculations also need edge corrections # The default edge correction in spatstat is generally the best, but can be slow, so we'll use the "border" correction for speed here # Uncertainties on K-function estimates can be assessed by randomly sampling points from a uniform Poisson process in the area and computing the K-function of the simulated data # Repeat this process 99 times, and take the minimum and maximum value of K over each of the distance values # This gives an envelope - if the K-function from the data goes above the top of the envelope then we have evidence for clustering # If the K-function goes below the envelope then there is evidence for an inhibitory process causing points to be spaced out # Envelopes can be computed using the envelope() function # The plot method for estimates of K uses a formula system where a dot on the left of a formula refers to K® # So the default plot uses . ~ r # You can compare the estimate of K to a Poisson process by plotting . - pi * r ^ 2 ~ r # If the data was generated by a Poisson process, then the line should be close to zero for all values of r # Point patterns are pre-defined p_poisson## Planar point pattern: 501 points ## window: polygonal boundary ## enclosing rectangle: [-10, 10] x [-10, 10] unitsp_cluster## Planar point pattern: 332 points ## window: polygonal boundary ## enclosing rectangle: [-10, 10] x [-10, 10] unitsp_regular## Planar point pattern: 325 points ## window: polygonal boundary ## enclosing rectangle: [-10, 10] x [-10, 10] units# Estimate the K-function for the Poisson points K_poisson <- Kest(p_poisson, correction = "border") # The default plot shows quadratic growth plot(K_poisson, . ~ r)
# Subtract pi * r ^ 2 from the Y-axis and plot plot(K_poisson, . - pi * r**2 ~ r)
# Compute envelopes of K under random locations K_cluster_env <- envelope(p_cluster, Kest, correction = "border")## Generating 99 simulations of CSR ... ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, ## 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, ## 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99. ## ## Done.# Insert the full formula to plot K minus pi * r^2 plot(K_cluster_env, . - pi * r^2 ~ r)
# Repeat for regular data K_regular_env <- envelope(p_regular, Kest, correction = "border")## Generating 99 simulations of CSR ... ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, ## 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, ## 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99. ## ## Done.plot(K_regular_env, . - pi * r^2 ~ r)
Chapter 2 - Point Pattern Analysis
Bivariate point problems:
- People tend to clump together in cities in neighborhoods, so “things people do/cause” (accidents, crime, etc.) are likely to be clumped even in the absence of differential rates
- The bivariate point pattern is the solution - look at both where things do happen and also where they do not happen (e.g., with disease, looking at prevalence rate rather than total patient counts by area)
- Can run either a bivariate nearest neighbors or a bivariate K-function
- The null hypothesis is usually one of uniform rates, which is to say that there is no clustering of incidents and controls
- Can run either a bivariate nearest neighbors or a bivariate K-function
- Kernel smoothing can help with plotting the rates - replace each event with a kernel, and assign every point the sum of all kernels
- Need to pick the shape and bandwidth of the kernel
- The spatstat package has some reasonable defaults, and these can be over-ridden if desired
- Need to pick the shape and bandwidth of the kernel
Spatial segregation:
- The null hypothesis is typically that rates are constant wherever you go within the window (though volumes might clump)
- The spseg() function estimates a bandwidth for the kernel function
- This is then used for creating the rates by point, which can be plotted
- The spseg() function estimates a bandwidth for the kernel function
- Monte Carlo simulation can be used to permute the events, keeping event locations and relative ratio of events within the window constant
- Can then find from simulation the 95% CI for event rate, and identify outliers on the graph
- Can use plotmc() somehow to get the confidence intervals overlayed on the original plot, although it does not always work directly with ppp
- Can then find from simulation the 95% CI for event rate, and identify outliers on the graph
Space-time data:
- Can have a purely temporal process, which also includes a “window” (start and end time)
- Each coordinate will now have a t coordinate, so frequently (x, y, t)
- The t needs to be numeric, so make a switch if it was included as a Character or Date or POSIXct or the like
- Can plot in several ways, though typically the time is shown as a histogram while the (x, y) are plotted as in previous chapters
- Each coordinate will now have a t coordinate, so frequently (x, y, t)
- The rggobi package can provide linked plot, where you brush point by time and see various highlighting associated to the selection
Space-time clustering:
- Often, there is interest in testing a hypothesis of space-time independence
- The Space-Time K Function typically has time on the y-axis and “spatial distance” on the x-axis
- Basically, there is a cylinder of space time, for points within distance d and also within time t of each other
- Assuming independence, then Ks(s) * Kt(t) = Kst(s, t)
- Can define a test statistic D(s, t) = Kst(s, t) - Ks(s) * Kt(t)
- Basically, there is a cylinder of space time, for points within distance d and also within time t of each other
- The test consists of Monte Carlo with permuted event times - same distances, same times, but not interaction effect
- Compare the observed test statistic D with the Monte Carlo null simulations
- Can run stmctest(myPoints, myTimes, shapeWindow, timeWindow, dIntervals, tIntervals, nSim)
- Compare the observed test statistic D with the Monte Carlo null simulations
Example code includes:
# The dataset we shall use for this example consists of crimes in a 4km radius of the center of Preston, a town in north-west England # We want to look for hotspots of violent crime in the area # A ppp object called preston_crime has been constructed # This is a marked point process, where each point is marked as either a "Violent Crime" or a "Non-violent crime" # The marks for each point can be retrieved using the marks() function # The window is a 4km circle centered on the town center # A map image of the town from OpenStreetMap has also been loaded, called preston_osm preston_crime <- readRDS("./RInputFiles/pcrime-spatstat.RDS") preston_osm <- readRDS("./RInputFiles/osm_preston_gray.RDS") # Get some summary information on the dataset summary(preston_crime)## Marked planar point pattern: 2036 points ## Average intensity 4.053214e-05 points per square unit ## ## Coordinates are given to 2 decimal places ## i.e. rounded to the nearest multiple of 0.01 units ## ## Multitype: ## frequency proportion intensity ## Non-violent crime 1812 0.8899804 3.607281e-05 ## Violent crime 224 0.1100196 4.459332e-06 ## ## Window: polygonal boundary ## single connected closed polygon with 99 vertices ## enclosing rectangle: [349773, 357771] x [425706.5, 433705.5] units ## Window area = 50231700 square units ## Fraction of frame area: 0.785# Get a table of marks table(marks(preston_crime))## ## Non-violent crime Violent crime ## 1812 224# Define a function to create a map preston_map <- function(cols = c("green","red"), cex = c(1, 1), pch = c(1, 1)) { raster::plotRGB(preston_osm) # from the raster package plot(preston_crime, cols = cols, pch = pch, cex = cex, add = TRUE, show.window = TRUE) } # Draw the map with colors, sizes and plot character preston_map( cols = c("black", "red"), cex = c(0.5, 1), pch = 19 )
# One method of computing a smooth intensity surface from a set of points is to use kernel smoothing # Imagine replacing each point with a dot of ink on absorbent paper # Each individual ink drop spreads out into a patch with a dark center, and multiple drops add together and make the paper even darker # With the right amount of ink in each drop, and with paper of the right absorbency, you can create a fair impression of the density of the original points # In kernel smoothing jargon, this means computing a bandwidth and using a particular kernel function # To get a smooth map of violent crimes proportion, we can estimate the intensity surface for violent and non-violent crimes, and take the ratio # To do this with the density() function in spatstat, we have to split the points according to the two values of the marks and then compute the ratio of the violent crime surface to the total # The function has sensible defaults for the kernel function and bandwidth to guarantee something that looks at least plausible # preston_crime has been pre-defined preston_crime## Marked planar point pattern: 2036 points ## Multitype, with levels = Non-violent crime, Violent crime ## window: polygonal boundary ## enclosing rectangle: [349773, 357771] x [425706.5, 433705.5] units# Use the split function to show the two point patterns crime_splits <- split(preston_crime) # Plot the split crime plot(crime_splits)
# Compute the densities of both sets of points crime_densities <- density(crime_splits) # Calc the violent density divided by the sum of both frac_violent_crime_density <- crime_densities[[2]] / (crime_densities[[1]] + crime_densities[[2]]) # Plot the density of the fraction of violent crime plot(frac_violent_crime_density)
# We can get a more principled measure of the violent crime ratio using a spatial segregation model # The spatialkernel package implements the theory of spatial segregation # The first step is to compute the optimal bandwidth for kernel smoothing under the segregation model # A small bandwidth would result in a density that is mostly zero, with spikes at the event locations # A large bandwidth would flatten out any structure in the events, resulting in a large "blob" across the whole window # Somewhere between these extremes is a bandwidth that best represents an underlying density for the process # spseg() will scan over a range of bandwidths and compute a test statistic using a cross-validation method # The bandwidth that maximizes this test statistic is the one to use # The returned value from spseg() in this case is a list, with h and cv elements giving the values of the statistic over the input h values # The spatialkernel package supplies a plotcv function to show how the test value varies # The hcv element has the value of the best bandwidth # spatstat is loaded and the preston_crime object is read in # Scan from 500m to 1000m in steps of 50m bw_choice <- spatialkernel::spseg( preston_crime, h = seq(500, 1000, by = 50), opt = 1)## ## Calculating cross-validated likelihood function# Plot the results and highlight the best bandwidth spatialkernel::plotcv(bw_choice) abline(v = bw_choice$hcv, lty = 2, col = "red")
# Print the best bandwidth print(bw_choice$hcv)## [1] 800# The second step is to compute the probabilities for violent and non-violent crimes as a smooth surface, as well as the p-values for a point-wise test of segregation # This is done by calling spseg() with opt = 3 and a fixed bandwidth parameter h # Normally you would run this process for at least 100 simulations, but that will take too long to run here # Instead, run for only 10 simulations # Then you can use a pre-loaded object seg which is the output from a 1000 simulation run that took about 20 minutes to complete # Set the correct bandwidth and run for 10 simulations only seg10 <- spatialkernel::spseg( pts = preston_crime, h = bw_choice$hcv, opt = 3, ntest = 10, proc = FALSE) # Plot the segregation map for violent crime spatialkernel::plotmc(seg10, "Violent crime")
# Plot seg, the result of running 1000 simulations (not included here) # spatialkernel::plotmc(seg, "Violent crime") # With a base map and some image and contour functions we can display both the probabilities and the significance tests over the area with more control than the plotmc() function. # The seg object is a list with several components # The X and Y coordinates of the grid are stored in the $gridx and $gridy elements # The probabilities of each class of data (violent or non-violent crime) are in a matrix element $p with a column for each class # The p-value of the significance test is in a similar matrix element called $stpvalue # Rearranging columns of these matrices into a grid of values can be done with R's matrix() function # From there you can construct list objects with a vector $x of X-coordinates, $y of Y-coordinates, and $z as the matrix # You can then feed this to image() or contour() for visualization # This process may seem complex, but remember that with R you can always write functions to perform complex tasks and those you may repeat often # For example, to help with the mapping in this exercise you will create a function that builds a map from four different items # The seg object from 1000 simulations is loaded, as well as the preston_crime points and the preston_osm map image # Inspect the structure of the spatial segregation object # str(seg) # Get the number of columns in the data so we can rearrange to a grid # ncol <- length(seg$gridx) # Rearrange the probability column into a grid # prob_violent <- list(x = seg$gridx, # y = seg$gridy, # z = matrix(seg$p[, "Violent crime"], # ncol = ncol)) # image(prob_violent) # Rearrange the p-values, but choose a p-value threshold # p_value <- list(x = seg$gridx, # y = seg$gridy, # z = matrix(seg$stpvalue[, "Violent crime"] < 0.05, # ncol = ncol)) # image(p_value) # Create a mapping function # segmap <- function(prob_list, pv_list, low, high){ # # # background map # plotRGB(preston_osm) # # # p-value areas # image(pv_list, # col = c("#00000000", "#FF808080"), add = TRUE) # # # probability contours # contour(prob_list, # levels = c(low, high), # col = c("#206020", "red"), # labels = c("Low", "High"), # add = TRUE) # # # boundary window # plot(Window(preston_crime), add = TRUE) # } # # # Map the probability and p-value # segmap(prob_violent, p_value, 0.05, 0.15) # The sasquatch, or "bigfoot", is a large ape-like creature reported to live in North American forests # The Bigfoot Field Researchers Organization maintains a database of sightings and allows its use for teaching and research # A cleaned subset of data in north-west USA has been created as the ppp object sasq and is loaded for you to explore the space-time pattern of sightings in the area # Get a quick summary of the dataset sasq <- readRDS("./RInputFiles/sasquatch.RDS") summary(sasq)## Marked planar point pattern: 423 points ## Average intensity 2.097156e-09 points per square unit ## ## *Pattern contains duplicated points* ## ## Coordinates are given to 1 decimal place ## i.e. rounded to the nearest multiple of 0.1 units ## ## Mark variables: date, year, month ## Summary: ## date year month ## Min. :1990-05-03 Y2004 : 41 Sep : 59 ## 1st Qu.:2000-04-30 Y2000 : 36 Oct : 56 ## Median :2003-11-05 Y2002 : 30 Aug : 54 ## Mean :2003-08-11 Y2005 : 30 Jul : 50 ## 3rd Qu.:2007-11-02 Y2001 : 26 Nov : 43 ## Max. :2016-04-05 Y2008 : 26 Jun : 41 ## (Other):234 (Other):120 ## ## Window: polygonal boundary ## single connected closed polygon with 64 vertices ## enclosing rectangle: [368187.8, 764535.6] x [4644873, 5434933] units ## Window area = 2.01702e+11 square units ## Fraction of frame area: 0.644# Plot unmarked points plot(unmark(sasq))
# Plot the points using a circle sized by date plot(sasq, which.marks = "date")
# Show the available marks names(marks(sasq))## [1] "date" "year" "month"# Histogram the dates of the sightings, grouped by year hist(marks(sasq)$date, "years", freq = TRUE)
# Plot and tabulate the calendar month of all the sightings plot(table(marks(sasq)$month))
# Split on the month mark sasq_by_month <- split(sasq, "month", un = TRUE) # Plot monthly maps plot(sasq_by_month)
# Plot smoothed versions of the above split maps plot(density(sasq_by_month))
# To do a space-time clustering test with stmctest() from the splancs package, you first need to convert parts of your ppp object # Functions in splancs tend to use matrix data instead of data frames. # To run stmctest() you need to set up # event locations # event times # region polygon # time limits # the time and space ranges for analysis # The sasq object is loaded and the spatstat and splancs packages are ready for use # Get a matrix of event coordinates sasq_xy <- as.matrix(coords(sasq)) # Check the matrix has two columns dim(sasq_xy)## [1] 423 2# Get a vector of event times sasq_t <- marks(sasq)$date # Extract a two-column matrix from the ppp object sasq_poly <- as.matrix(as.data.frame(Window(sasq))) dim(sasq_poly)## [1] 64 2# Set the time limit to 1 day before and 1 day after the range of times tlimits <- range(sasq_t) + c(-1, 1) # Scan over 400m intervals from 100m to 20km s <- seq(100, 20000, by = 400) # Scan over 14 day intervals from one week to 31 weeks tm <- seq(7, 217, by = 14) # Everything is now ready for you to run the space-time clustering test function # You can then plot the results and compute a p-value for rejecting the null hypothesis of no space-time clustering # Any space-time clustering in a data set will be removed if you randomly rearrange the dates of the data points # The stmctest() function computes a clustering test statistic for your data based on the space-time K-function - how many points are within a spatial and temporal window of a point of the data # It then does a number of random rearrangements of the dates among the points and computes the clustering statistic # After doing this a large number of times, you can compare the test statistic for your data with the values from the random data # If the test statistic for your data is sufficiently large or small, you can reject the null hypothesis of no space-time clustering # The output from stmctest() is a list with a single t0 which is the test statistic for your data, and a vector of t from the simulations # By converting to data frame you can feed this to ggplot functions # Because the window area is a large number of square meters, and we have about 400 events, the numerical value of the intensity is a very small number # This makes values of the various K-functions very large numbers, since they are proportional to the inverse of the intensity # Don't worry if you see 10^10 or higher # The p-value of a Monte-Carlo test like this is just the proportion of test statistics that are larger than the value from the data # You can compute this from the t and t0 elements of the output # All the objects from the previous exercise are loaded. # Run 999 simulations sasq_mc <- splancs::stmctest(sasq_xy, sasq_t, sasq_poly, tlimits, s, tm, nsim = 999, quiet = TRUE) names(sasq_mc)## [1] "t0" "t"# Histogram the simulated statistics and add a line at the data value ggplot(data.frame(sasq_mc), aes(x = t)) + geom_histogram(binwidth = 1e13) + geom_vline(aes(xintercept = t0))
# Compute the p-value as the proportion of tests greater than the data sum(sasq_mc$t > sasq_mc$t0) / 1000## [1] 0.04
Chapter 3 - Areal Statistics
Areal statistics:
- Areal statistics is the idea of point data being aggregated in to a region (for example, for confidentiality reasons)
- Boundaries are typically note created in a manner that supports good statistical analysis
- Boundaries are typically note created in a manner that supports good statistical analysis
- The cartogram is one solution to areal statistics - the cartogram makes the area proportional to a unit of interest (e.g., population)
- The simplest hypothesis is that data are spatially random, which is to say that each region is indepenent of all the other regions
- The alternative hypothesis is that “neighbors” tend to be more similar than random regions
- Can run packages to get the neighbors for each region, frequently defined similar to neighbors for a graph (network)
- Can also convert the list resulting from the neighbors package to an adjacency matrix of 1/0 for who is neighbors
- The alternative hypothesis is that “neighbors” tend to be more similar than random regions
- The Moran I statistic is a test statistic that can be used to assess similarities or differences among neighbors
- Large positive values of I mean that neighbors tend to be more similar
- Large negative values of I mean that neighbors tend to be more dissimilar
- Near-zero values of I mean that neighbors tend to be no more or less similar than any random regions
- Monte Carlo tests by permuting regional values can build a null distribution
- Large positive values of I mean that neighbors tend to be more similar
Spatial health data:
- Population health data is frequently available open-source, and often comes in standard forms
- There is sometimes a challenge in getting the proper rates since the denominator should be “at risk” and not “total population”
- Incidence Rate is the Number of Cases divided by Population at Risk (typically between 0 and 1)
- Common to standardize incidence rate as Incidence-Rate-Local / Incidence-Rate-Global, known as SMR (standardized morbidity ratio)
- The expected number of cases in a region would be Incidence-Rate-Global * Regional-AtRisk-Population (can also be converted to SMR)
- There is sometimes a challenge in getting the proper rates since the denominator should be “at risk” and not “total population”
- Can run hypothesis tests and confidence intervals on the SMR
- Can also shade maps based on SMR values (e.g., red for SMR > 2 with 99%+ probability)
Generalized linear models in space:
- Count data are naturally modeled by the Poisson distribution where Y ~ Poisson(e ** (X %*% Beta))
- When applying GLM to spatial data, can have a map of residuals
- The residuals should be “uncorrelated” and look random
- If the residuals are “correlated”, then inferences about the parameter estimates are very possibly mistaken
- There are techniques for dealing with spatial correlations of the residuals
- The residuals should be “uncorrelated” and look random
Correlation in spatial GLM:
- Check whether the residuals can be explained by another known factor - add them to the model as needed
- An additional approach is to explcitly add a spatial term S(x, y) to the model
- The conditional autocorrelation model says that any given region should have mean equal to the average of its neighbors, with variance to be fitted
- The “car” model stands for “conditionally autocorrelated regression”
- The conditional autocorrelation model says that any given region should have mean equal to the average of its neighbors, with variance to be fitted
- Bayesian statistics are frequently used, basically, what are the likelihoods for the parameter given the observed data?
- The credible interval is the Bayesian equivalent for confidence interval
Example code includes:
library(cartogram) library(rgeos)## rgeos version: 0.3-26, (SVN revision 560) ## GEOS runtime version: 3.6.1-CAPI-1.10.1 r0 ## Linking to sp version: 1.2-7 ## Polygon checking: TRUElibrary(spdep)## Loading required package: sp## Loading required package: Matrix## ## Attaching package: 'Matrix'## The following object is masked from 'package:tidyr': ## ## expand## Loading required package: spData## To access larger datasets in this package, install the spDataLarge ## package with: `install.packages('spDataLarge')`## ## Attaching package: 'spData'## The following objects are masked _by_ '.GlobalEnv': ## ## x, ylibrary(epitools) library(R2BayesX)## Loading required package: BayesXsrc## Loading required package: colorspace## ## Attaching package: 'colorspace'## The following object is masked from 'package:spatstat': ## ## coords## Loading required package: mgcv## This is mgcv 1.8-17. For overview type 'help("mgcv-package")'.# In 2016 the UK held a public vote on whether to remain in the European Union # The results of the referendum, where people voted either "Remain" or "Leave", are available online # The data set london_ref contains the results for the 32 boroughs of London, and includes the number and percentage of votes in each category as well as the count of spoilt votes, the population size and the electorate size # The london_ref object is a SpatialPolygonsDataFrame, a special kind of data frame where each row also has the shape of the borough # It behaves like a data frame in many respects, but can also be used to plot a choropleth, or shaded polygon, map # You should start with some simple data exploration and mapping. The following variables will be useful: # NAME : the name of the borough. # Electorate : the total number of people who can vote. # Remain, Leave : the number of votes for "Remain" or "Leave". # Pct_Remain, Pct_Leave : the percentage of votes for each sid # spplot() from the raster package provides a convenient way to draw a shaded map of regions # See what information we have for each borough london_ref <- readRDS("./RInputFiles/london_eu.RDS") summary(london_ref)## Object of class SpatialPolygonsDataFrame ## Coordinates: ## min max ## x 503574.2 561956.7 ## y 155850.8 200933.6 ## Is projected: TRUE ## proj4string : ## [+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 ## +y_0=-100000 +datum=OSGB36 +units=m +no_defs +ellps=airy ## +towgs84=446.448,-125.157,542.060,0.1502,0.2470,0.8421,-20.4894] ## Data attributes: ## NAME TOTAL_POP Electorate Votes_Cast ## Length:32 Min. :157711 Min. : 83042 Min. : 54801 ## Class :character 1st Qu.:237717 1st Qu.:143458 1st Qu.:104079 ## Mode :character Median :272017 Median :168394 Median :116280 ## Mean :270780 Mean :169337 Mean :118025 ## 3rd Qu.:316911 3rd Qu.:196285 3rd Qu.:134142 ## Max. :379691 Max. :245349 Max. :182570 ## Remain Leave Rejected_Ballots Pct_Remain ## Min. : 27750 Min. :17138 Min. : 60.0 Min. :30.34 ## 1st Qu.: 55973 1st Qu.:32138 1st Qu.:105.0 1st Qu.:53.69 ## Median : 70254 Median :45263 Median :138.0 Median :61.01 ## Mean : 70631 Mean :47255 Mean :139.0 Mean :60.46 ## 3rd Qu.: 84287 3rd Qu.:59018 3rd Qu.:164.2 3rd Qu.:69.90 ## Max. :118463 Max. :96885 Max. :267.0 Max. :78.62 ## Pct_Leave Pct_Rejected Assembly ## Min. :21.38 Min. :0.0600 Length:32 ## 1st Qu.:30.10 1st Qu.:0.0875 Class :character ## Median :38.99 Median :0.1100 Mode :character ## Mean :39.54 Mean :0.1187 ## 3rd Qu.:46.31 3rd Qu.:0.1500 ## Max. :69.66 Max. :0.2200# Which boroughs voted to "Leave"? london_ref$NAME[london_ref$Leave > london_ref$Remain]## [1] "Sutton" "Barking and Dagenham" "Bexley" ## [4] "Havering" "Hillingdon"# Plot a map of the percentage that voted "Remain" sp::spplot(london_ref, zcol = "Pct_Remain")
# Large areas, such as cities or countries, are often divided into smaller administrative units, often into zones of approximately equal population # But the area of those units may vary considerably # When mapping them, the large areas carry more visual "weight" than small areas, although just as many people live in the small areas. # One technique for correcting for this is the cartogram # This is a controlled distortion of the regions, expanding some and contracting others, so that the area of each region is proportional to a desired quantity, such as the population # The cartogram also tries to maintain the correct geography as much as possible, by keeping regions in roughly the same place relative to each other # The cartogram package contains functions for creating cartograms # You give it a spatial data frame and the name of a column, and you get back a similar data frame but with regions distorted so that the region area is proportional to the column value of the regions # You'll also use the rgeos package for computing the areas of individual regions with the gArea() function # Use the cartogram and rgeos packages (called at top of routine) # library(cartogram) # library(rgeos) # Make a scatterplot of electorate vs borough area names(london_ref)## [1] "NAME" "TOTAL_POP" "Electorate" ## [4] "Votes_Cast" "Remain" "Leave" ## [7] "Rejected_Ballots" "Pct_Remain" "Pct_Leave" ## [10] "Pct_Rejected" "Assembly"plot(london_ref$Electorate, gArea(london_ref, byid = TRUE))
# Make a cartogram, scaling the area to the electorate carto_ref <- cartogram(london_ref, "Electorate")## Mean size error for iteration 1: 1.5881743190908## Mean size error for iteration 2: 1.32100446455657## Mean size error for iteration 3: 1.18227723476121## Mean size error for iteration 4: 1.10676057030171## Mean size error for iteration 5: 1.0657703107641## Mean size error for iteration 6: 1.04259259672006## Mean size error for iteration 7: 1.02832326230708## Mean size error for iteration 8: 1.01931941526112## Mean size error for iteration 9: 1.01341424685212## Mean size error for iteration 10: 1.00941370606418## Mean size error for iteration 11: 1.00663364742297## Mean size error for iteration 12: 1.00470553629914## Mean size error for iteration 13: 1.00336434720465## Mean size error for iteration 14: 1.00241457265516## Mean size error for iteration 15: 1.00174179254187plot(carto_ref)
# Check the linearity of the electorate-area plot plot(carto_ref$Electorate, gArea(carto_ref, byid = TRUE))
# Make a fairer map of the Remain percentage sp::spplot(carto_ref, "Pct_Remain")
# The map of "Remain" votes seems to have spatial correlation # Pick any two boroughs that are neighbors - with a shared border - and the chances are they'll be more similar than any two random boroughs # This can be a problem when using statistical models that assume, conditional on the model, that the data points are independent # The spdep package has functions for measures of spatial correlation, also known as spatial dependency # Computing these measures first requires you to work out which regions are neighbors via the poly2nb() function, short for "polygons to neighbors" # The result is an object of class nb # Then you can compute the test statistic and run a significance test on the null hypothesis of no spatial correlation # The significance test can either be done by Monte-Carlo or theoretical models # In this example you'll use the Moran "I" statistic to test the spatial correlation of the population and the percentage "Remain" vote. # The london_ref spatial data object is loaded for you # Use the spdep package (called at top of routine) # library(spdep) # Make neighbor list borough_nb <- poly2nb(london_ref) # Get center points of each borough borough_centers <- coordinates(london_ref) # Show the connections plot(london_ref) plot(borough_nb, borough_centers, add = TRUE)
# Map the total pop'n sp::spplot(london_ref, zcol = "TOTAL_POP")
# Run a Moran I test on total pop'n moran.test( london_ref$TOTAL_POP, nb2listw(borough_nb) )## ## Moran I test under randomisation ## ## data: london_ref$TOTAL_POP ## weights: nb2listw(borough_nb) ## ## Moran I statistic standard deviate = 1.2124, p-value = 0.1127 ## alternative hypothesis: greater ## sample estimates: ## Moran I statistic Expectation Variance ## 0.11549264 -0.03225806 0.01485190# Map % Remain sp::spplot(london_ref, zcol = "Pct_Remain")# Run a Moran I MC test on % Remain moran.mc( london_ref$Pct_Remain, nb2listw(borough_nb), nsim = 999 )## ## Monte-Carlo simulation of Moran I ## ## data: london_ref$Pct_Remain ## weights: nb2listw(borough_nb) ## number of simulations + 1: 1000 ## ## statistic = 0.42841, observed rank = 1000, p-value = 0.001 ## alternative hypothesis: greater# The UK's National Health Service publishes weekly data for consultations at a number of "sentinel" clinics and makes this data available # A dataset for one week in February 2017 has been loaded for you to analyze # It is called london, and contains data for the 32 boroughs. # You will focus on reports of "Influenza-like illness", or more simply "Flu" # Your first task is to map the "Standardized Morbidity Ratio", or SMR # This is the number of cases per person, but scaled by the overall incidence so that the expected number is 1 # The london object, a spatial data frame, and the sp package are ready for you # Get a summary of the data set london <- readRDS("./RInputFiles/london_2017_2.RDS") summary(london)## Object of class SpatialPolygonsDataFrame ## Coordinates: ## min max ## x 503574.2 561956.7 ## y 155850.8 200933.6 ## Is projected: TRUE ## proj4string : ## [+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 ## +y_0=-100000 +datum=OSGB36 +units=m +no_defs +ellps=airy ## +towgs84=446.448,-125.157,542.060,0.1502,0.2470,0.8421,-20.4894] ## Data attributes: ## CODE NAME Flu_OBS Vom_OBS ## Length:32 Length:32 Min. : 0.00 Min. : 0.00 ## Class :character Class :character 1st Qu.: 11.00 1st Qu.:14.00 ## Mode :character Mode :character Median : 33.00 Median :40.00 ## Mean : 38.56 Mean :37.28 ## 3rd Qu.: 61.00 3rd Qu.:57.50 ## Max. :112.00 Max. :96.00 ## Diarr_OBS Gastro_OBS TOTAL_POP CCGcode ## Min. : 0.00 Min. : 0.0 Min. :157711 Length:32 ## 1st Qu.: 22.50 1st Qu.: 48.0 1st Qu.:237717 Class :character ## Median : 62.00 Median :120.5 Median :272017 Mode :character ## Mean : 57.03 Mean :113.7 Mean :270780 ## 3rd Qu.: 90.75 3rd Qu.:176.8 3rd Qu.:316911 ## Max. :122.00 Max. :251.0 Max. :379691 ## CCG.geography.code CCG.name Asthma_Prevalence ## Length:32 Length:32 Min. :3.550 ## Class :character Class :character 1st Qu.:4.412 ## Mode :character Mode :character Median :4.660 ## Mean :4.624 ## 3rd Qu.:4.925 ## Max. :5.470 ## Obesity_Prevalence Cancer_Prevalence Diabetes_Prevalence Income ## Min. : 3.930 Min. :0.870 Min. :3.620 Min. :0.0730 ## 1st Qu.: 5.855 1st Qu.:1.438 1st Qu.:5.265 1st Qu.:0.1308 ## Median : 7.565 Median :1.605 Median :6.305 Median :0.1665 ## Mean : 7.585 Mean :1.684 Mean :6.245 Mean :0.1655 ## 3rd Qu.: 8.810 3rd Qu.:1.903 3rd Qu.:7.067 3rd Qu.:0.1985 ## Max. :12.130 Max. :2.540 Max. :9.060 Max. :0.2530 ## Employment Education HealthDeprivation Crime ## Min. :0.0570 Min. : 3.958 Min. :-1.4100 Min. :-0.1550 ## 1st Qu.:0.0905 1st Qu.:10.047 1st Qu.:-0.5055 1st Qu.: 0.3745 ## Median :0.1095 Median :13.925 Median :-0.2050 Median : 0.5515 ## Mean :0.1092 Mean :14.024 Mean :-0.2044 Mean : 0.5379 ## 3rd Qu.:0.1283 3rd Qu.:17.480 3rd Qu.: 0.2010 3rd Qu.: 0.7823 ## Max. :0.1560 Max. :27.182 Max. : 0.5430 Max. : 1.0190 ## Services Environment i ## Min. :19.63 Min. :13.37 Min. : 0.00 ## 1st Qu.:24.43 1st Qu.:24.03 1st Qu.: 7.75 ## Median :30.41 Median :28.20 Median :15.50 ## Mean :29.55 Mean :31.38 Mean :15.50 ## 3rd Qu.:34.74 3rd Qu.:40.15 3rd Qu.:23.25 ## Max. :41.89 Max. :55.00 Max. :31.00# Map the OBServed number of flu reports sp::spplot(london, "Flu_OBS")
# Compute and print the overall incidence of flu r <- sum(london$Flu_OBS) / sum(london$TOTAL_POP) r## [1] 0.0001424128# Calculate the expected number for each borough london$Flu_EXP <- london$TOTAL_POP * r # Calculate the ratio of OBServed to EXPected london$Flu_SMR <- london$Flu_OBS / london$Flu_EXP # Map the SMR sp::spplot(london, "Flu_SMR")
# SMRs above 1 represent high rates of disease - but how high does an SMR need to be before it can be considered statistically significant? # Given a number of cases and a population, its possible to work out confidence intervals at some level of the estimate of the ratio of cases per population using the properties of the binomial distribution # The epitools package has a function binom.exact() which you can use to compute confidence intervals for the flu data # These can be scaled to be confidence intervals on the SMR by dividing by the overall rate # The london data set and the sp package are loaded # For the binomial statistics function (called at top of routine) # library(epitools) # Get CI from binomial distribution flu_ci <- binom.exact(london$Flu_OBS, london$TOTAL_POP) # Add borough names flu_ci$NAME <- london$NAME # Calculate London rate, then compute SMR r <- sum(london$Flu_OBS) / sum(london$TOTAL_POP) flu_ci$SMR <- flu_ci$proportion / r # Subset the high SMR data flu_high <- flu_ci[flu_ci$SMR > 1, ] # Plot estimates with CIs ggplot(flu_high, aes(x = NAME, y = proportion / r, ymin = lower / r, ymax = upper / r)) + geom_pointrange() + theme(axis.text.x = element_text(angle = 45, hjust = 1))
# Distributions and confidence intervals can be difficult things to present to non-statisticians # An alternative is to present a probability that a value is over a threshold # For example, public health teams might be interested in when an SMR has more than doubled, and as a statistician you can give a probability that this has happened # Then the public health team might decide to go to some alert level when the probability of a doubling of SMR is over 0.95 # Again, the properties of the binomial distribution let you compute this for proportional data # You can then map these exceedence probabilities for some threshold, and use a sensible color scheme to highlight probabilities close to 1 # The london data set has been loaded, and the expected flu case count, Flu_EXP has been computed # Probability of a binomial exceeding a multiple binom.exceed <- function(observed, population, expected, e){ 1 - pbinom(e * expected, population, prob = observed / population) } # Compute P(rate > 2) london$Flu_gt_2 <- binom.exceed( observed = london$Flu_OBS, population = london$TOTAL_POP, expected = london$Flu_EXP, e = 2) # Use a 50-color palette that only starts changing at around 0.9 pal <- c( rep("#B0D0B0", 40), colorRampPalette(c("#B0D0B0", "orange"))(5), colorRampPalette(c("orange", "red"))(5) ) # Plot the P(rate > 2) map sp::spplot(london, "Flu_gt_2", col.regions = pal, at = seq(0, 1, len = 50))
# A Poisson generalized linear model is a way of fitting count data to explanatory variables # You get out parameter estimates and standard errors for your explanatory variables, and can get fitted values and residuals # The glm() function fits Poisson GLMs. It works just like the lm() function, but you also specify a family argument # The formula has the usual meaning - response on the left of the ~, and explanatory variables on the right # To cope with count data coming from populations of different sizes, you specify an offset argument # This adds a constant term for each row of the data in the model. The log of the population is used in the offset term # The london health data set has been loaded with the sp package also ready. # Run a Poisson generalized linear model of how the count of flu cases, Flu_OBS, depends on the Health Deprivation index value, HealthDeprivation # The first argument is the formula (response vairable on the left) # The family argument is a function, poisson # Fit a poisson GLM. model_flu <- glm( Flu_OBS ~ HealthDeprivation, offset = log(TOTAL_POP), data = london, family = "poisson") # Is HealthDeprivation significant? summary(model_flu)## ## Call: ## glm(formula = Flu_OBS ~ HealthDeprivation, family = "poisson", ## data = london, offset = log(TOTAL_POP)) ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -9.5361 -4.5285 -0.0499 2.9043 8.2194 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) -8.78190 0.02869 -306.043 <2e-16 *** ## HealthDeprivation 0.65689 0.06797 9.665 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for poisson family taken to be 1) ## ## Null deviance: 703.75 on 31 degrees of freedom ## Residual deviance: 605.03 on 30 degrees of freedom ## AIC: 762.37 ## ## Number of Fisher Scoring iterations: 5# Put residuals into the spatial data. london$Flu_Resid <- residuals(model_flu) # Map the residuals using spplot sp::spplot(london, "Flu_Resid")
# A linear model should fit the data and leave uncorrelated residuals # This applies to non-spatial models, where, for example, fitting a straight line through points on a curve would lead to serially-correlated residuals # A model on spatial data should aim to have residuals that show no significant spatial correlation # You can test the model fitted to the flu data using moran.mc() from the spdep package # Monte Carlo Moran tests were previously discussed in the Spatial autocorrelation test exercise earlier in the chapter # Compute the neighborhood structure. borough_nb <- poly2nb(london) # Test spatial correlation of the residuals. moran.mc(london$Flu_Resid, listw = nb2listw(borough_nb), nsim = 999)## ## Monte-Carlo simulation of Moran I ## ## data: london$Flu_Resid ## weights: nb2listw(borough_nb) ## number of simulations + 1: 1000 ## ## statistic = 0.15059, observed rank = 925, p-value = 0.075 ## alternative hypothesis: greater# Bayesian statistical models return samples of the parameters of interest (the "posterior" distribution) based on some "prior" distribution which is then updated by the data # The Bayesian modelling process returns a number of samples from which you can compute the mean, or an exceedence probability, or any other quantity you might compute from a distribution # Before you fit a model with spatial correlation, you'll first fit the same model as before, but using Bayesian inference # The london data set has been loaded # The R2BayesX package provides an interface to the BayesX code. # The syntax for bayesx() is similar, but the offset has to be specified explicitly from the data frame, the family name is in quotes, and the spatial data frame needs to be turned into a plain data frame # Run the model fitting and inspect with summary() # Plot the samples from the Bayesian model # On the left is the "trace" of samples in sequential order, and on the right is the parameter density # For this model there is an intercept and a slope for the Health Deprivation score # The parameter density should correspond with the parameter summary # Use R2BayesX (called at top of routine) # library(R2BayesX) # Fit a GLM model_flu <- glm(Flu_OBS ~ HealthDeprivation, offset = log(TOTAL_POP), data = london, family = poisson) # Summarize it summary(model_flu)## ## Call: ## glm(formula = Flu_OBS ~ HealthDeprivation, family = poisson, ## data = london, offset = log(TOTAL_POP)) ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -9.5361 -4.5285 -0.0499 2.9043 8.2194 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) -8.78190 0.02869 -306.043 <2e-16 *** ## HealthDeprivation 0.65689 0.06797 9.665 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for poisson family taken to be 1) ## ## Null deviance: 703.75 on 31 degrees of freedom ## Residual deviance: 605.03 on 30 degrees of freedom ## AIC: 762.37 ## ## Number of Fisher Scoring iterations: 5# Calculate coeff confidence intervals confint(model_flu)## Waiting for profiling to be done...## 2.5 % 97.5 % ## (Intercept) -8.838677 -8.7261843 ## HealthDeprivation 0.524437 0.7908841# Fit a Bayesian GLM bayes_flu <- bayesx(Flu_OBS ~ HealthDeprivation, offset = log(london$TOTAL_POP), family = "poisson", data = as.data.frame(london), control = bayesx.control(seed = 17610407)) # Summarize it summary(bayes_flu)## Call: ## bayesx(formula = Flu_OBS ~ HealthDeprivation, data = as.data.frame(london), ## offset = log(london$TOTAL_POP), control = bayesx.control(seed = 17610407), ## family = "poisson") ## ## Fixed effects estimation results: ## ## Parametric coefficients: ## Mean Sd 2.5% 50% 97.5% ## (Intercept) -8.7831 0.0278 -8.8371 -8.7841 -8.7263 ## HealthDeprivation 0.6592 0.0659 0.5345 0.6587 0.7900 ## ## N = 32 burnin = 2000 method = MCMC family = poisson ## iterations = 12000 step = 10# Look at the samples from the Bayesian model plot(samples(bayes_flu))
# You've fitted a non-spatial GLM with BayesX # You can include a spatially correlated term based on the adjacency structure by adding a term to the formula specifying a spatially correlated model # Use poly2nb() to compute the neighborhood structure of london to an nb object # R2BayesX uses its own objects for the adjacency. Convert the nb object to a gra object # The sx function specifies additional terms to bayesx. Create a term using the "spatial" basis and the gra object for the boroughs to define the map # Print a summary of the model object. You should see a table of coefficients for the parametric part of the model as in the previous exercise, and then a table of "Smooth terms variance" with one row for the spatial term # Note that since BayesX can fit many different forms in its sx terms, most of which, like the spatial model here, cannot be simply expressed with a parameter or two # This table shows the variance of the random effects - for further explanation consult a text on random effects modelling # Compute adjacency objects borough_nb <- poly2nb(london) borough_gra <- nb2gra(borough_nb) # Fit spatial model flu_spatial <- bayesx( Flu_OBS ~ HealthDeprivation + sx(i, bs = "spatial", map = borough_gra), offset = log(london$TOTAL_POP), family = "poisson", data = data.frame(london), control = bayesx.control(seed = 17610407) )## Note: created new output directory 'C:/Users/Dave/AppData/Local/Temp/Rtmp4cxBim/bayesx1'!# Summarize the model summary(flu_spatial)## Call: ## bayesx(formula = Flu_OBS ~ HealthDeprivation + sx(i, bs = "spatial", ## map = borough_gra), data = data.frame(london), offset = log(london$TOTAL_POP), ## control = bayesx.control(seed = 17610407), family = "poisson") ## ## Fixed effects estimation results: ## ## Parametric coefficients: ## Mean Sd 2.5% 50% 97.5% ## (Intercept) -9.2311 0.1246 -9.4826 -9.2298 -9.0148 ## HealthDeprivation 0.7683 0.5844 -0.3749 0.7775 1.7922 ## ## Smooth terms variances: ## Mean Sd 2.5% 50% 97.5% Min Max ## sx(i):mrf 4.6381 1.6822 2.2851 4.3510 8.8104 1.6557 16.266 ## ## N = 32 burnin = 2000 method = MCMC family = poisson ## iterations = 12000 step = 10# As with glm, you can get the fitted values and residuals from your model using the fitted and residuals functions. bayesx models are a bit more complex, since you have the linear predictor and terms from sx elements, such as the spatially correlated term # The summary function will show you information for the linear model terms and the smoothing terms in two separate tables # The spatial term is called "sx(i):mrf" - standing for "Markov Random Field" # Bayesian analysis returns samples from a distribution for our S(x) term at each of the London boroughs # The fitted function from bayesx models returns summary statistics for each borough # You'll just look at the mean of that distribution for now # The model from the BayesX output is available as flu_spatial. # Summarise the model summary(flu_spatial)## Call: ## bayesx(formula = Flu_OBS ~ HealthDeprivation + sx(i, bs = "spatial", ## map = borough_gra), data = data.frame(london), offset = log(london$TOTAL_POP), ## control = bayesx.control(seed = 17610407), family = "poisson") ## ## Fixed effects estimation results: ## ## Parametric coefficients: ## Mean Sd 2.5% 50% 97.5% ## (Intercept) -9.2311 0.1246 -9.4826 -9.2298 -9.0148 ## HealthDeprivation 0.7683 0.5844 -0.3749 0.7775 1.7922 ## ## Smooth terms variances: ## Mean Sd 2.5% 50% 97.5% Min Max ## sx(i):mrf 4.6381 1.6822 2.2851 4.3510 8.8104 1.6557 16.266 ## ## N = 32 burnin = 2000 method = MCMC family = poisson ## iterations = 12000 step = 10# Map the fitted spatial term only london$spatial <- fitted(flu_spatial, term = "sx(i):mrf")[, "Mean"] sp::spplot(london, zcol = "spatial")
# Map the residuals london$spatial_resid <- residuals(flu_spatial)[, "mu"] sp::spplot(london, zcol = "spatial_resid")
# Test residuals for spatial correlation moran.mc(london$spatial_resid, nb2listw(borough_nb), 999)## ## Monte-Carlo simulation of Moran I ## ## data: london$spatial_resid ## weights: nb2listw(borough_nb) ## number of simulations + 1: 1000 ## ## statistic = -0.26922, observed rank = 16, p-value = 0.984 ## alternative hypothesis: greater
Chapter 4 - Geostatistics
- Each row of the body is